Theorems and Counterexamples in Mathematics (Problem Books in Mathematics)

q; denote the set of all
Sp(A) denote EnEP linj r(A) denote {S,p(A) :

(2.1.3.1)

Exercise 2.1.3.7. Show that if a = N and V = IR then the series Sa(A) ~f S(A) , i.e., E:'=l an, is absolutely convergent, i.e., 00

SII(A) ~f

L lanl <

00,

n=l

iff anyone (and hence each) of the following obtains: i. for each permutation (bijection)

7r :

N3 n

1-+

7r( n)

E N of N the series

00

S.r(A) ~f

L a.r(n) n=l

converges to a (real) number independent of 7rj

ii. r(A) is a bounded subset of IRj iii. for each E ~f {fn}nEN, fn = ±1, the series SE(A) ~f

E:'=l fna n

converges. If SI I(A) = 00 and S(A) converges (S(A) is conditionally convergent), e.g., if an = (-I)n~, n E N, then r(A) is unbounded both above and below.

Exercise 2.1.3.8. Show that if S(A) is conditionally convergent then: i. for each closed interval J ~f [p, qj there is in II a permutation

7r

such

that the set E.r(A) of partial sums sn(7r)(A) ~f E~=l a.r(k),n E N, is such that the set E.r(A)' of limit points of E.r(A) is Jj ii. for each closed interval J there is a sequence E ~f {fn } nEN, fn = ± 1, such that SE(A)' = J. Note the special cases: p = q and p = -00, q = corresponds for i to the Riemann derangement theorem:

00.

The latter

Chapter 2. Analysis

70

IF SeA) IS CONDITIONALLY CONVERGENT, AND IF IS IN II A 7r SUCH THAT x = S.,..(A).

x E IR THERE

[Hint: The series formed of the positive terms and the series formed of the negative terms are divergent and limn -+ co an = o. If -00 < p < q < 00 choose 7r so that the terms in a subsequence of the partial sums in S.,.. are alternately below p and above q. If f > 0 there is in Nan n(f) such that if n > n(f) and sn(7r)(A) < p and sn+k(7r)(A) > q then ISn+j+l(7r)(A) - Sn+j(7r)(A)1 < f, j = 0, 1, ... ,k - 1. If -00 = p < q < 00 choose 7r so that for n in N, the partial sums are alternately below -n and above q. Similar constructions serve for the circumstances -00 < p < q = 00 and -00 = p < q = 00. The same kinds of arguments apply for SE(A).] Let Z

=

def {

Zn

=

def ( ) }

Znl, ... , Znr

nEN

• be a sequence of vectors m V

=

def

In [St] Steinitz proved the following generalization of the Riemann derangement theorem. THEOREM 2.1.3.1. IF S(Z) IS A SERIES OF VECTORS IN A FINITEDIMENSIONAL VECTOR SPACE V THEN THE SET

SueZ) ~f {S.,..(Z) :

7r

Ell}

OF ALL CONVERGENT SUMS OF REARRANGEMENTS OF S(Z) IS EMPTY OR THERE IS A SUBSPACE M AND A VECTOR X SUCH THAT

SueZ)

[ Note 2.1.3.1:

If M

= x+M.

= {O} then Su = x.]

[Remark 2.1.3.2: The Steinitz paper, published in 1913 but written in 1906, provides a beautiful but somewhat old-fashioned introduction to the subject of vector spaces. In [Rosn] there is a simplified proof of the Steinitz theorem.] Sierpinski [Sil] proved the following version of the Riemann derangement theorem. THEOREM

2.1.3.2.

IF ~:=l

an

IS A CONDITIONALLY CONVERGENT

s' :$ S ~f ~:=l an THEN an < 0 ::} a.,..(n) = an AND ~:=l a,..(n) = s'.

SERIES IN WHICH THE TERMS ARE REAL AND IF THERE IS IN

II

A

7r

SUCH THAT

[Remark 2.1.3.3: In [Sil] the author refers to an earlier paper he wrote in his native language, Polish. In that paper he showed

Section 2.1. Classical Real Analysis

71

that if a series of real terms is conditionally convergent and if s is an arbitrary real number there is a rearrangement of the series in which the positions of the + and - signs are those of the original series and yet the sum of the series is s.] Exercise 2.1.3.9. Let {zn ~f (c n , dn)}nEN be a sequence of vectors in R2. Use THEOREM 2.1.3.1. 70 as needed to show:

i. if SII II ~f L~=lllznll = 00 then L~=llcnl = 00 or L~=lldnl = 00; ii. if SII II = 00, if S ~f L~=l Zn converges (in R2) (S is conditionally convergent), and if L~=llcnl < 00 then {S.,.. : 7r E II, S.,.. converges} is the set L~=l Cn + iR; iii. (x, y) denoting the inner product of the vectors x and y, if a) x ~f (Xl, X2) and y d~f (Yl. Y2) are linearly independent, b) (x, S.,..) converges for every 7r in II, c) (y, S) is conditionally convergent, then (x, S.,..) is a constant K independent of 7r, each (convergent) S.,.. is on the line: L ~f {z : (x, z) = K}, and for each A in R there is in II a 7r such that S.,.. converges and is on the line: def L>. = {w : (W,Y)=A, AER},

i.e., Sn

= L.

[Hint: Show that M in THEOREM 2.1.3.1. 70 is one-dimensional.] If ak = 2- k , kEN, then an :5 L~n+l ak, n E N, and if there is in N a subset Ax such that LnEAz an = x.

Exercise 2.1.3.10.

then for every LnEAz an

X

= x.

in

a) Show that if

(0, L~=l an ~f s]

b) Show that if 00

N, and for every x in

°<

an

X

E (0,1]

:5 L~=n+l ak <

00

there is in N a subset Ax for which

> S = L~=l an > L~=l an > 0,

(0, L~=l an ~f S]

N E

there is in N a subset Ax such that

x = LnEAz an, then for some permutation 7r of N, a.,..(n) :5 L~=n+l a.,..(k)· [Hint: ad a) By induction define a sequence {nk} so that each nk is the smallest n such that anI + ... + a nk _ 1 + an :5 x. ad b)

°

Consider a permutation 7r such that a.,..(n) ~f bn ! and show that for each n in N, 1 bl + ... + bn+l :5 2 (S + bl + ... + bn ) .] The classical Cauchy criterion for convergence of a series S(A) of vectors in a vector space V complete with respect to the norm II II is, because

Chapter 2. Analysis

72

it is a criterion, not always simple to apply. Hence, in the study of series in which the terms are constants (real or complex) or not necessarily constant functions, there has evolved a collection of useful tests for convergence. Among the tests most frequently applied is the comparison test in the following form: If S(A) ~f E:=l an and S(B) ~f E:=l bn are series in which the terms are nonnegative and if S(A) dominates S(B), i.e., if an ~ bn , then the convergence of S(A) implies the convergence of S(B) (whence the divergence of S(B) implies the divergence of

S(A)). For the applicability of comparison test the condition that the terms of the series under study be nonnegative is crucial. If the terms are not necessarily nonnegative, i.e., if some may be negative, the comparison test may fail to be decisive. A series S(A) is said to dominate S(B) absolutely if an ~ Ibnl, n E N, (whence the terms of S(A) are nonnegative, although the terms of S(B) may be positive, zero, or negative). Exercise 2.1.3.11. Give an example of a divergent series such that limn ..... oo an = O.

E:=l an

Exercise 2.1.3.12. Give an example of a convergent series S(A) and a divergent series S(B) such that an ~ bn . Exercise 2.1.3.13. Give an example of a convergent series S(A) and a divergent series S(B) for which lanl ~ Ibnl, n EN. [Hint: In the last two Exercises the solution must involve series in which some terms are negative.] A series S(A) of positive terms would provide a universal comparison test for all series with positive terms if S(A) dominated absolutely every convergent series with positive terms, and S(A) were dominated absolutely by every divergent series with positive terms. However if S(A) diverges then S(A) is not dominated absolutely by the divergent series ~S(A): S(A) fails to test ~S(A) by comparison. If S(A) converges and then S(A) does not dominate absolutely the convergent series 2S(A): S(A) fails to test 2S(A) by comparison. There is no single series S(A) that can serve as a universal comparison test. As THEOREM 2.1.3.3 below shows, even more is true. THEOREM 2.1.3.3. LET 5 ~f {E:=l amn}nEN ~f {S(An)}nEN BE A SEQUENCE OF CONVERGENT SERIES OF POSITIVE TERMS. THEN THERE IS A CONVERGENT SERIES S(A) OF POSITIVE TERMS THAT IS NOT DOMINATED BY ANY MEMBER OF S.

Section 2.1. Classical Real Analysis

73

[Remark 2.1.3.4: The result above may be paraphrased briefly by saying that there is no universal comparison sequence of positive convergent series.] PROOF. If SM(An) ~f L~=l a mn and RM(An) = S(An) - SM(An) there are natural numbers M(n) such that 1:5 M(I) < M(2) < ... and

Let am be if 1 :5 m :5 M(2) if M(k) < m :5 M(k

{ 2aml (k + 1) max(amb . .. ,amk)

+ 1),

k

> 1, kEN.

Then S(A) ~f L:'=l am converges since M(k+l) [ L am :5 L (k m=M(k)+l m=M(k)+l

k

M(Hl)

:5 (k + 1)

+ 1) L

1

amn

n=l

t[

Mfl)

a mn ]

m=M(k)+l

n=l k

:5 (k + 1) L RM(k) (An) n=l k

:5 (k + 1) L 2- k < (k + 1)22-k. n=l

Hence M(2)

00

S(A) ~f L

m=l

am

=L

(M(k+l) ) L am k=2 m=M(k)+l 00

am

+L

m=l

00

:5 SM(2) (Ad

+ L(k + 1)22- k < 00. k=2

Since am/a mn ~ (k + 1) if k ~ nand m > M(k) it follows that the convergent series S(A) is not dominated by any KS(An), K E JR.

o The idea of the preceding proof can be used to show that there is no sequence of positive divergent series that serves as a universal comparison series sequence for divergence.

74

Chapter 2. Analysis

Even when a series diverges some generalized averaging method might lead to a "reasonable" value to assign as the sum of the series. Such a generalized averaging method is often termed a summability method that is used to sum the series.

= E:'=l(-I)n+l then

Example 2.1.3.1. If S(A)

sn(A) = {01 if n is odd if n is even. It follows that the average

(Tn

(A)
+ ... + 8 n(A) _

{~

-!

n

if n is odd if n is even.

!, which is regarded as an acceptable

Hence lim n-+ oo (Tn (A) = the divergent series S(A).

"value" of

Exercise 2.1.3.14. Let I be Lebesgue integrable on [-11",11"], and let the nth Fourier coefficient of I be Cn

1 111" . rn= l(x)e- mX dx, n E Z.

def

=

v211"

-11"

Let Sf(x) be the (formal) Fourier series

L 00

n=-oo N

in:z:

cn _e _

~

inz

and let SN(X) be En=-N Cn $. Show that the average N

(TN(X)

=

def

" sn(X) N 1+ 1 'L..J

n=O

-111" 1 [sin!(N+l)(x-y)/sin!(x-y)]2 () -2 N I l y dy. -11" 11" + The function F (x N

) ,y

~ ~ [sin !(N + l)(x - y)/ sin !(x _ y)]2 211"

N

+1

is Fejer's kernel and the integral in the second line of the display above is the convolution, denoted FN * I, of FN and I.

Section 2.1. Classical Real Analysis

75

In discussing convergence resp. uniform convergence it is helpful to use the notation fn -+ g resp. fn ~ g to signify that the sequence {fn}nEN converges resp. converges uniformly to g as n -+ 00. Among the properties of Fejer's kernel are [Zy]: i. FN 2:: OJ ii. f::1r FN(X} dx = Ij m. if 0 < f < 7r then FN ~ 0 in [-7r,7r] \ (-f,f).

Exercise 2.1.3.15. Show: i. the validity of i-iii above for FNj ii. that if f is continuous on [-7r,7r] and f(-7r}

= f(7r}

then

(Fejer's theorem)j iii. that if f is Lebesgue integrable on [-7r,7r] then

IIf - fN

* fill ~f

i:

If(x) - FN

[Hint: For iii use ii and the fact that if 8 function g such that IIf - gill < 8.]

* f(x}1 dx -+ O.

> 0 there is a continuous

More generally let T ~f {t mn }:,n=l be a (Toeplitz) matrix in which each entry is real and for which: i. ii. ... au.

there is an M such that E~lltmnl ~ M, mE N'j limm..... oo tmn = 0, n E N'j I'Imm ..... oo ",,00 t 1 L.m=l mn = . If O'm,T(A} ~f E:=l tmnsn(A} converges for m in N' and if

exists then O'T(A} is the T-sum of S(A}. Thus if

t mn -_{-ml

o

ifl~n~m otherwise

!,

then the T-sum of E:=l(-I}n+l is i.e., T sums S(A}. A matrix T is a Toeplitz matrix iff whenever S(A} converges then its T-sum is also S(A}: O'T(A} = S(A} [To, Wi].

Chapter 2. Analysis

76

Exercise 2.1.3.16. Show that if tmn

= {~ o

if 1 ~ n ~ m otherwise

then T ~f (tmn)~::'l is a Toeplitz matrix and corresponds to the simple averaging procedure described above, cf. (2.1.3.2), page 79. There are two large classes of Toeplitz matrices, those derived from Cesaro summation, denoted (C,a), and those derived from Abel summation. Details about the following statements are discussed in [Zy]. i. If

a>

-1, 0

< x < 1, A = {an}~=o

n Sn

= Lak,

n EN

k=O

ISnl S

~ M

< 00, n E N ,,",00

n

(A) ~f L.tn-O anx x,a (1 _ x)a+l

s~xn) (~f r::=o (1- x)a

and if

. s~ def hm -A = s(C a)(A) ~ ,

n ...... oo

exists then S(A) is said to be (C,a)-summable to 8(C,a)(A). ii. If lim Sx o(A) ~f SAbel (A) z ...... 1

'

exists then S(A) is Abel-summable to SAbel(A). iii. If -1 < a < (3 a) and S(C,a)(A) exists then S(A) is (C,(3)-summable (S(c,p)(A) exists) and S(c,p)(A) = S(C,a)(A); b) there is an A such that S(A) is (C,(3)-summable but is not (C,a)summable; c) and if S(A) is (C,a)-summable it is Abel-summable; d) there is an A such that, for each a in (-1,00), S(A) is not (C,a)summable but is Abel-summable. iv. For each a in (-1,00) there is a Toeplitz matrix Ta such that lim U m T", m-+oo' whenever S(C,a)(A) exists.

(A) = S(C ,a) (A)

Section 2.1. Classical Real Analysis

77

v. If Xn II and x ~f {Xn}nEN there is a Toeplitz matrix Tx such that lim

n-+oo

Um

'

Tx(A) = SAbel(A)

whenever SAbel(A) exists.

Example 2.1.3.1. 74 shows there are divergent series that can be summed by some Toeplitz matrices. Exercise 2.1.3.17. Show that the (infinite) identity matrix I ~f (!5ij)f,j'=l is a Toeplitz matrix that sums a series SeA) iff SeA) converges. There is no "universal" Toeplitz matrix that sums every series. More emphatic is the next result (cf. THEOREM 2.1.3.3. 72). THEOREM 2.1.3.4. LET {T(k)hEN BE A COUNTABLE SET OF TOEPLITZ MATRICES. THEN THERE IS A SERIES SeA) SUCH THAT

EXISTS FOR EACH m AND EACH k. YET FOR EACH k IN N, lim

m ...... oo

U

m

T(k)

(A)

t

DOES NOT EXIST, [GeO, Hab].

PROOF. Assume that T(k) = {t~l}:,n=l' Owing to i-iii in the definition of a Toeplitz matrix, there are in N two strictly increasing sequences: {mp}PEN and {nphEN so that: if m ~ m1 and k = 1, 00

L t~l = 1 + f1m,

If1ml < 0.05,

n=l

00

nl

L t~~n = 1 +!5

b

11511

< 2(0.05),

n=l ifm

~

L n=nl

It~~nl < 0.05; +1

m2 and k = 1,2, 00

L t~l = 1 + f2m,

n=l

nl

If2ml < (0.05)2,

L It~ll < (0.05)2,

n=l

n2

00

n=l

n=n2+1

L t~~n = 1 + 152, 11521< 2(0.05)2, L

It~~nl < (0.05)2;

Chapter 2. Analysis

78

and, in general, if m

~

m p , k = 1,2, ... ,p, and pEN, np-l

00

L t~~ = 1 + fpm,

Ifpml

< (0.05)P,

n=l np

L t~~n = 1 + op,

L It~~1 < (0.05)P,

n=l 00

lopl

< 2(0.05)P,

L

It~~nl < (0.05)p.

n=l Let S(A) be such that

sn(A)

~f

{I

if 1 :5 n :5 nl. n2 < n :5 n3, .. . if nl < n :5 n2, n3 < n :5 n4, .. ..

-1

(The sequence A itself can be calculated according to the formula

an = If p is odd, p Um

pt

T(k)

> 1,

{ Sl(A) sn(A) - sn-l(A)

and 1

ifn=1 if 1 < n EN.)

:5 k :5 p then

nl (A) = "L...J t(k ) mpR n=l

n2

-

" L...J

t(k) mpn

+ ...

n=nl+l 00

- ... + "L...J

_ ... +

t(k) mpn Sn (A)

(~t(k) _ ~l t(k) ) + ~ L...J~n L...J ~n L...J n=l

n=l

t(k) S

~nn

(A).

n=np+l

The conditions imposed on the sequences {mp}PEN and {np}pEN imply that Ump,T(k)

(A)

> 1 - 2(0.05)P-l - 2(p - 1)(0.05)P-l - (0.05)P (= 1 - [2(P - 1)

+ 2.05] (0.05)P-l ~f f(p))

.

Since f'(p) = (0.05)p-l [-2 - [2(p - 1) + 2.05]lnO.05j it follows that if pis odd and p > 1 then f' (p) > 0 and thus on [3, 00) the minimum value of f(p) is f(3). Hence if p is odd and p > 1 then Ump,T(k)

(A)

> 0.9.

A similar argument shows that if p is even then U

m p, T(k) (A)

< -0.7.

Section 2.1. Classical Real Analysis

79

Therefore for each k in N the sequence

{Un,T(k) (A)}nEN

does not converge.

D The formula e Z = lim (1 n-+oo

+ =')n n

may be related to the Toeplitz matrix

o o o 1 o o 1 1 1 3 3 3 o 1 1

'T'

.LC,l

def

=

(2.1.3.2)

cf. Exercise 2.1.3.16. 76, corresponding to averaging the terms of a sequence. The formula has a generalization in terms of Toeplitz matrices. Exercise 2.1.3.18. Assume that T ~f (tij)~'~ is a Toeplitz matrix such that

Show that if z E C then 00

.lim II(1

1-+00

+ tijZ) = e

Z•

j=l

Give an example of a Toeplitz matrix for which conclusion above is not valid.

it

[Hint: If 0 < 6 < there is a constant K and a function ai(z) such that if Izi < 6 then lai(z)1 :5 K and

The next Exercises illustrate some of the unexpected phenomena in the study of series. Exercise 2.1.3.19. The alternating series theorem states that if, for nE N,

i. En = (_1)n+l u. an ~ an+1 iii. an ! 0

Chapter 2. Analysis

80

then E:'l fna n converges. Show that each series below diverges and that for it only the indicated alternating series condition is violated:

f

n=l

~ (i)

f(-l)n n=l

nn~od2

(ii)

00

I)-l)n (iii). n=l

Exercise 2.1.3.20. Show that if bn

> 0, n EN, and liminfb n =0 n-+oo

there is a divergent series S(A) in which the terms are positive, lim an n~oo

= 0,

and lim inf abn n--+oo

n

= O.

[Remark 2.1.3.5: Hence, no matter how rapidly the positive sequence B ~f {bn}nEN converges to 0 there is a positive sequence A ~f {an}nEN converging to 0 so slowly that S(A) diverges and yet A contains a subsequence converging to 0 more rapidly than the corresponding B-subsequence.] [Hint: Choose a sequence {nkhEN such that nl + 1 < ... and such that

< nl + 1 < n2 <

n2

lim bnk

k-+oo

= O.

Choose an so that

= nk, kEN if n = mj EN \ {nl. n2,"'} ~f {ml. m2," if n

.}.]

Exercise 2.1.3.21. For a given positive sequence B such that liminfbn n-+oo

=0

Section 2.1. Classical Real Analysis

81

find a positive sequence A such that S(A) converges and yet an . 11m sup -b n~oo

n

= 00.

What is the counterpart of Remark 2.1.3.5. 80? [Hint: There is a sequence N ~f {nkhEN such that bnk Choose an so that

_{*"

an -

fi2'

<

k- 3 •

ifn = nk if n ¢ N.]

In the next Exercises dealing with ratio and root tests all series considered are assumed to have positive terms. Exercise 2.1.3.22. Show that although the limiting ratio exists the ratio test fails for: 1

00

L

n2

00

(convergent) and

n=l

1

L;i' (divergent). n=l

Exercise 2.1.3.23. Show that the limiting ratio does not exist and the generalized ratio test for S(A) i. limsuPn-+ oo ii. limsuPn-+ oo

a:!l a:!l

< 1 => convergence > 1 => divergence

fails for 00

00

n=l

n=l

L 2(-l)"-n (convergent) and L 2n-(-1)n (divergent). Exercise 2.1.3.24. Show that the generalized root test for S(A) .1.

i. lim sUPn-+oo a;;

.1.

ii. lim sUPn-+oo a;;

< 1 => convergence > 1 => divergence

fails for

L..J n~=1(5+(2-1)n)-n

00 (convergent) and ~

(5+(2- l)n)n (divergent).

Chapter 2. Analysis

82

Exercise 2.1.3.25. For a given positive sequence A show that · . fa n +l . f a,~J. IHum - < _ I·Hum n~oo

an

n~oo

< an+l. _ I·un sup a;:J. < _ I·1m sup n~oo

n~oo

an

[Remark 2.1.3.6: Hence the (generalized) ratio test can conceivably fail while the (generalized) root test succeeds.] Exercise 2.1.3.26. Show that the root test succeeds (while the generalized ratio test fails) for 00

L

00

2(-1)" -n

(convergent) and

n=1

L2

n -(-I)"

(divergent).

n=1

The Mertens theorem [01] states that if one of S(A) and S(B) converges absolutely and both converge then their Cauchy product n

00

S(C) ~f

L(L

00

akbn-k+l)

~f

n=1 k=1

L

Cn

n=1

converges to S(A)S(B). Exercise 2.1.3.27. Show that if A = B = {(-1)n(n+1)-!}nEN then S(A) (hence S(B)) converges but that their Cauchy product does not. [Hint: Show that since vi(l + x)(n + 1 - x) achieves its maximum on [0, n] when x

=~

it follows that

Ien I > -

2(n + 1) ] n+2 .

Exercise 2.1.3.28. Show that the Cauchy product of the divergent series 00

2+

L

00

2n

and - 1 +

n=2

L1

n

n=2

converges. Most of the material above deals with series of constants. In the next discussion the emphasis is on series of terms that are not necessarily constants. Exercise 2.1.3.29. Show that S(A, x) ~f E~o e- n cos n 2 x represents a function f in Coo, that the Maclaurin series for f consists only of terms of even degree, and the absolute value of the term of degree 2k is (

oc

~

e -n n 4k) 2k (2k)! x (>

(

n2x ) 2k

2k

-n

e), n EN.

Section 2.1. Classical Real Analysis

83

Show that if x i' 0 and k > 2~ then the term of degree 2k is greater than 1, whence that the Maclaurin series for I converges iff x = o.

Example 2.1.3.2. Assume

and, by means of bridging functions, rPnO is made infinitely differentiable everywhere and 0 ::5 rPno (x) ::5 ((n - 1)!f If

then direct calculation shows that (k)

lIn (x)l::5

(2- n +l) Ixl n - k - 2 n2 (n _ k - 2)!

whence the Weierstmft M -test shows that for all k in {OJ UN,

converges uniformly on every finite interval. It follows that 00

I(x) ~f

L In(x) E Coo n=l

and that its Maclaurin series is 00

Ln!xn , n=O

which converges iff x series

= O.

A similar argument shows that an arbitrary

is the Maclaurin series of some function in Coo.

Chapter 2. Analysis

84

[Remark 2.1.3.7: The function character from that of

f just described is different in if x ~ 0 otherwise

(cf. (2.1.2.2), page 61). The Maclaurin series for g converges everywhere and represents g only at o. The Maclaurin series for f converges only at 0 (and represents f only at 0).] Associated with the power series P(x) ~f E:=o anx n is the number L ~f limsuPn-+oo lanl~. The radius of convergence R of P(x) is given by the formula:

R={i

~fL>O

00 If L

= o.

If 0 ~ r < R then P(x) converges uniformly and absolutely in the interval [-r, r]. There is no general result about uniformity of convergence in (-R,R) nor about convergence if x = ±R (when R < 00).

Example 2.1.3.3. The radii of convergence R}, R2, R3 for the power series n=1

are all 1. Yet: i. PI(x) converges uniformly in any closed subinterval of (-1,1), converges but not uniformly in (-1, 1), and diverges if x = ± 1; ii. P2 (x) behaves just like PI(x) except that P2 (x) converges if x = -1; iii. P3 (x) converges uniformly on [-1, 1].

Thus the opportunities for finding abnormality of convergence behavior are somewhat limited if the domain of study is the set of power series. On the other hand, orthonormal series, in particular trigonometric series, and more particularly Fourier series provide many examples of unusual convergence phenomena. [Remark 2.1.3.8: It was the study of trigonometric series that gave rise to the proper definition by Riemann of the integral bearing his name. It was the study of the sets of convergence and divergence of trigonometric series that led Cantor to the study of

Section 2.1. Classical Real Analysis

85

"sets" and thereby opened the field of modern set theory, logic, cardinal and ordinal numbers, etc.] Let f be integrable on [-11', 11']. The Riemann-Lebesgue theorem implies that the sequence Cn

1 111' f () = . tn= x e- i n v211' -11'

def

:J:

dx, n E Z

of Fourier coefficients of f converges to 0 as Inl for f is written in the form

-+ 00.

If the Fourier series

1 00 2'ao + L(ancosnx+bnsinnx) n=l

then for n in N, an

=k

(cn

lim (a!

n-+oo

+ c-n ),

bn

=

*

(cn - c-n ). Hence

+ b!) ~£ n-+oo lim p! = o.

Cantor, whose research preceded the development of the modern theory of (Lebesgue) integration, showed that if the trigonometric series 1 2'ao

00

+ Lan cosnx + bn sinnx n=l

(which need not be the Fourier series of an integrable function, cf. Example 2.1.3.5. 87) is such that lim an cos nx + bn sin nx ~f lim Pn cos(nx + an)

n-+oo

n-+oo

=0

(2.1.3.3)

everywhere then liffin-+oo Pn = O. Lebesgue sharpened Cantor's result as follows. THEOREM 2.1.3.5. (CANTOR-LEBESGUE) LET (2.1.3.3) OBTAIN EVERYWHERE ON A MEASURABLE SET E OF POSITIVE MEASURE. THEN limn -+oo Pn = O. PROOF. If Pn f+ 0 as n -+ 00 then there is a sequence {nkhEN and a positive f such that Pnk ~ f for all k. Hence limn -+ oo cos( nkX + ant) = 0 a.e. on E and the left member of

tends to 0 as k -+ 00. The Riemann-Lebesgue theorem implies that the second term in the right member above tends to 0 as k -+ 00 and it follows that ~A(E) = 0, a contradiction. 0

Chapter 2. Analysis

86

It should be noted that although the orthogonal set T of trigonometric functions is complete in £2 ([0,211'], C), the completeness of T plays no role in the validity of the last result. Indeed, the argument remains accurate if only a proper but infinite subset of T is at hand. There follow variations inspired by the theme above. THEOREM 2.1.3.6. FOR THE MEASURE SITUATION (X,S,p.) LET {fn}nEN BE AN INFINITE ORTHONORMAL SET CONSISTING OF UNIFORMLY BOUNDED FUNCTIONS: IIfnlloo ~ M < 00, n EN. IF anfn(x) CONVERGES TO ZERO ALMOST EVERYWHERE THEN limn ..... oo an = O. PROOF. If the conclusion is false, via subsequences as needed, it may be assumed that for some positive f and each n in N, lanl ~ f. Hence fn(x)~'O and Ifn(x)1 2 ~. O. The bounded convergence theorem implies the contradiction 1 == Ifn(x)1 2 dp.(x) -+ O.

Ix

D On the other hand, absent the condition c) (uniform boundedness of the functions), the conclusion above may fail to obtain.

Example 2.1.3.4. Let En be (n~l' ~], n E N and for n in N, let fn be XEn' Then {gn ~f ";n(n + l)fn}nEN is an orthonormal set in £2([0,1], A) and for every sequence {an}nEN, E~=l angn converges a.e. [ Note 2.1.3.2: The particular "real" form

1 '2ao

00

+ Lan cosnx + bn sinnx n=l

of the trigonometric series gives the Cantor-Lebesgue theorem its significance. If a trigonometric series has the form inz

00

L n=-oo

Cn

ern= y211'

y;;;.

and if limlnl ..... oo Cn = 0 for even one value of x then, since leinzl == 1, it follows without further proof that limlnl ..... oo Cn = 0.] The next lemma, due to Abel, is useful in many arguments. LEMMA 2.1.3.1. IF S(A) IS A SERIES, IF bn ! 0, AND IF THERE IS AN M SUCH THAT ISn(A)1 ~ M, n E N THEN E~=l bnan CONVERGES [Kno]. The Euler formula (e±it ;....

~smnx

n=l

= cos t ± i sin t) _ cos ~ - cos

-

implies that

(N + ~) x

2 . z sm 2'

Section 2.1. Classical Real Analysis and so

87

tsinnxl :51~1· In=l SIn 2

If sin ~

1= 0, i.e.,if x is fixed and is not an integral multiple of 211', then 18i~ f I for all N in N. Abel's lemma now implies that if

I E:=l sin nxl

:5

bn ! 0 then E~=l bn sin nx converges for any x that is not a multiple of 211'. Inspection shows that the series converges if x is a multiple of 211', whence the series converges everywhere. Example 2.1.3.5. Abel's lemma implies that the trigonometric series 00

•

E

smnx n=2 Inn

(2.1.3.4)

converges for all x. Example 2.1.3.6. The trigonometric series (2.1.3.4) is not the Fourier series of a Lebesgue integrable function. Indeed, if f(x) ~£

E smnx, 00

•

n=2 Inn

if f is Lebesgue integrable, and F(x)

~£ 1:11 f(t) dt,

then F is absolutely continuous, periodic (because f is), and even (because f is odd) whence the Fourier series for F is a cosine series that converges everywhere to F: F(x) if n ~ 2 then

= E~oancosnx. a

Integration by parts shows that

1

n

= -nlnn --

Thus if f is Lebesgue integrable there emerges the contradiction that the divergent series 00 1 ' " --cosnO ~ nlnn

00

1

= E-n=2 nlnn

converges: f represented by (2.1.3.4) is not Lebesgue integrable. In particular, although (2.1.3.4) converges everywhere it does not converge uniformly on [-11',11']. There remains the question of whether some Lebesgue integrable function 9 is such that its Fourier coefficients are those in (2.1.3.4). However, for such a 9 it follows from Exercise 2.1.3.15. 75 that IIg - FN * gilt -+ 0

Chapter 2. Analysis

88

as n --+ 00, whence, for some subsequence {NkhEN, FNk *g~' 9 as k --+ 00. Since the functions FNk * 9 are, as well, the averages of the partial sums of the series for / it follows that / = 9 a.e. Since / is not Lebesgue integrable no such 9 exists. [Remark 2.1.3.9: In particular (2.1.3.4) is not the Fourier series of a Riemann integrable function.]

Exercise 2.1.3.30. Show that

L smnnx 00

•

(2.1.3.5)

n=l

is the Fourier series for / : [-11',11'] :3 x 1-+ x, which is of bounded variation on [-11',11']: / E BV([-1I', 11']). Hence (2.1.3.5) converges uniformly on every closed subinterval of ( -11', 11') [Zy, I, p. 57]. Show, on the other hand, that the WeierstraB M-test is not effectively applicable: there is no convergent series of positive constants that dominates (2.1.3.5) on [a, b] if -11' < a < b < 11'. According to the Riemann-Lebesgue theorem the Fourier transform j(t)

~f _1_

.,fi/i

1

00

/(x)e- itz dx

-00

of a function / that is Lebesgue integrable on IR is a continuous function that vanishes at infinity: limltl ..... oo j(t) = O. The next result shows that not every continuous function vanishing at infinity is a Fourier transform of a Lebesgue integrable function.

Example 2.1.3.7. Let L~=-oo en exp(inx)/.,fi/i be the complex exponential form of (2.1.3.4), page 87 or of any trigonometric series converging everywhere to a function that is not Lebesgue integrable. If h is in Coo, h(x) = 0, if x ¢ and sUPzEIR Ih(x)1 = h(O) = 211' then

[-1,1],

00

L

cnh(x - n) ~f /(x)

(2.1.3.6)

n=-oo

is a series in which, for each x, only one term, namely the term for which x ~ n ~ x + is nonzero. It follows that the series represents a function / in Coo. For a given x, if x ~ n ~ x + then 1/(x)1 ~ 211'Ienl whence limlzl ..... oo /(x) = 0, i.e., / vanishes at infinity. On the other hand, if F is Lebesgue integrable on IR then

-1

1,

-1

f: 11r

m=-oo

-w

IF(t

+ 211'm) Idt ~

1

1

00

-00

IF(t)1 dt < 00.

Section 2.1. Classical Real Analysis Hence E:'=-oo IF(x

+ 211'm) I < 00

89 a.e.,

00

L

g(x) ~f

F(x + 211'm)

m=-oo is defined a.e., g(x + 211')

= g(x) a.e., and

i:

Ig(t)1 dt <

00.

If, to boot, P = f then a direct calculation shows that nth Fourier coefficient of g is Cn , a contradiction, since the Cn are not the Fourier coefficients of any Lebesgue integrable function. In [Zy] there is a wealth of information about Fourier series. In particular there are included details about the following counterexamples in Fourier series (cf. also Note 2.3.1.1. 158). i. [Fejer and Lebesgue, I, pp. 300-1]. If E is a countable subset of [-11', 11'] there is a (Lebesgue integrable) function f such that the Fourier series for f diverges on E and converges on [-11',11'] \ E. H. [Kolmogorov, I, pp. 305,310]. There is a Lebesgue integrable function f for which the Fourier series diverges everywhere. The technique of proof for the two results above and for many other theorems in the subject of Fourier series is the definition of a sequence {¢n}nEN of trigonometric polynomials and a sequence {an}nEN of constants so that the series

(2.1.3.7) converges in an appropriate sense and defines a function f with the desired properties. For i the two sequences are chosen so that (2.1.3.7) converges uniformly, whence f is continuous. For ii the two sequences are chosen so that the absolute values of the partial sums of (2.1.3.7) are uniformly bounded and the (2.1.3.7) converges a.e., whence f is defined a.e. and is integrable. [ Note 2.1.3.3:

Despite the result in ii the equation

J~oo

i:

If(x) - FN

* f(x)1 dx =

0

is valid. Consequently there is a subsequence {Nk hEN such that limk--+oo FNk * f(x) = f(x) a.e.]

Chapter 2. Analysis

90

Exercise 2.1.3.31. Show that if In(x)

= {O~

if x E Q otherwise

then In ~ O. (The In are discontinuous everywhere, their uniform limit is continuous (constant).)

Exercise 2.1.3.32. Show that if In(x) = (sinnx)/vn then In ~ 0 while I~ diverges everywhere. [Hint: If x =F 0 then Icos 2nxl = 1- cos 2 nx. Hence if Icos nxl < then Icos 2nxl >

!

!.]

Exercise 2.1.3.33. Show that if In(x) =

{lo'nf{n,~)

if 0 < x :5 1 otherwise

then each In is bounded on [0,1] but lim In(x) = n ..... oo

{~

if 0 < x :5 1 otherwise.

0

Hence the nonuniform limit of bounded functions can be unbounded.

Exercise 2.1.3.34. Show that if

f (x)

= {inf(1,nX)

sup(-1,nx)

n

if x ~ 0 if x < 0

then each In is continuous on lR. and def

n~~/n(x)=E(x)=

{1 -1

ifx~O

ifx
Hence the discontinuous function E(x) is the nonuniform limit of the continuous functions In. Repeat for gn(x) ~f xn, n EN, x E [0,1].

Exercise 2.1.3.35. Show that if max

f.(x) oJ:!

{

(~, ~ + 2n2 (x - ~) )

~ax (;.; -

2n2 (x -

(l?q -~, l?]q 'f xE (l?q'q+~ l? I)

I'f X

m

E

I

I

otherwise

1 :5 q < n, 0:5 p :5 q I(x) ~f {~ U

if x = ~, p, q E Z, q > 0, and (p, q) otherwIse

=1

Section 2.1. Classical Real Analysis

91

(see Figure 2.1.3.1) then In ~ In+b each In is continuous, and In -+ I. Hence I, which is continuous only on [0, 1] \ Q is the nonuniform limit of the continuous functions In.

y-axis 1

1

2 1

'3 1

1 5

4

o

A

1

1

4

3

1

2

2

3

'3

Figure 2.1.3.1. The graph of y

4

1 x-axis

= Is(x) on [0,1].

[Remark 2.1.3.10: In [My] it is noted that each In is continuous, positive everywhere, and if h n = then each h n is continuous and the limit of {hn(X)}nEN on [0,1] is

i,

{n 00

ifm,n E N, (m,n) otherwise.

= 1, and x = ~

The limit of {hn}nEN is finite only on (Q n [0, 1]) \ {O}, an Fu. In [My] there is the following generalization. Let S be a subset of [0, 1]. There is a sequence {hn}nEN such that each h n is continuous on [0,1] and if XES if x E ([0,1] \ S)

· h n ( X ).IS {finite 11m • fi 't In nl e

n-oo

iff S is an Fu .] Exercise 2.1.3.36. Show that the In in Exercise 2.1.3.34. 90 are such that lim [1 In(x) dx = [1 lim In (x) dx n-oo

10

10

n-oo

Chapter 2. Analysis

92 even though the convergence is not uniform.

Exercise 2.1.3.37. Show that if

f-

if 0 ::; x ::; if...!... <x<'1 ~n - n if-<x<1 n then

1

1

1

lim n-oo

Jor

1

0

In{x) dx

= -2

lim In (x) dx = O. n-oo

Hence the nonuniform limit of Riemann integrable functions can be Riemann integrable but its integral can fail to be the limit of the integrals of the approximating functions. Show that the functions In and I of Exercise 2.1.3.35. 90 exhibit the same failure. Uniform convergence of a sequence of Riemann integrable functions is a sufficient but not necessary condition for the Riemann integrability of the limit function.

Exercise 2.1.3.38. Let {In}nEN be the set of disjoint open intervals deleted from [0,1] in the construction of a Cantor-like set C for which A{C) is positive. If N = 2n - 1,2 ::; n E N, define IN so that it is piecewise linear, continuous, and if x E I n , 1 ::; n ::; N if x = 0 or x = 1 if x is the midpoint of an interval separating two adjacent intervals Jp and J q , 1 ::; p, q ::; N. Show:

i. each IN is continuous, bounded, and hence Riemann integrable; n. limN_oo IN ~f I exists and Discont(f) :J C; iii. I is bounded and not Riemann integrable. Hence on [0,1], I is the nonuniform limit of uniformly bounded, continuous (hence Riemann and Lebesgue integrable) functions and I is Lebesgue integrable but not Riemann integrable.

Exercise 2.1.3.39. Show that the terms of the series n

00

L

n=l

X

n

Section 2.1. Classical Real Analysis

93

converge uniformly to 0 on [0,1) but that the series itself does not converge uniformly on [0,1). Exercise 2.1.3.40. Show that if

f

(x) n

~f

{in

~f n ~s odd If n IS even

then limn_ oo In(x) = 0, x E JR, the convergence to 0 is not uniform, and yet 12m ~ 0 on JR. Dini's theorem [HeSt] states that a monotone sequence of continuous functions converging to a continuous function on a compact set converges uniformly. Exercise 2.1.3.41. Show that each of the following sequences offunctions in JRIR fails to conform to one of the italicized conditions in Dini's theorem and fails to converge uniformly.

f (x) ~f {O ~f x n

F~ x ~ 1 ' n E N

= 0 or IlfO<x
In (x) ~f x n , n EN, 0 ~ x ~ 1 if 0 < x < 21 '\ if ~~ < x ~ n ' n E N.

2n2x df In(x) ~ { n - 2n2 (x - 2~)

o

ifn<x~1

Exercise 2.1.3.42. Show that if x In(x) = (1 + n 2x 2) ,

Ixl ~ 1,

then In ~ 0 on [-1, I] but that iff x =F 0, I~(x)

n EN -+

0 as n

-+ 00.

Exercise 2.1.3.43. Show that if In(x) = x n , x E [0,1) n E N, then In ~ 0 on every closed subinterval J of [0,1) but that the convergence In -+ 0 on [0,1) is not uniform. Exercise 2.1.3.44. Show that if In(x)

= {~ ~f 0 ~ x 2~ n 2 o If x> n

then In ~ 0 on [0, (0) but 1000 In (x) dx

t

00

as n

-+ 00.

Example 2.1.3.8. There is in Q a sequence {an}nEN such that if I E Co ([0, 1]) ~f {I : IE JR[O,ll, IE C ([0,1]), 1(0)

=0}

Chapter 2. Analysis

94

then there is in N U {O} a sequence

°=

no

{nkhENU{O}

such that

< n1 < ... < nk < ...

The Stone- Weierstrafl theorem [BeSt, Loo, Sto2, Sto3] implies that for each m in N the set 'Pm ~f Q [xm] of polynomials in xm and with rational coefficients is II lloo-dense in Co ([0, 1]). If {fn}nEN is a countable II lloo-dense set in Co ([0, 1]) there can be defined by induction a sequence {Pn}nEN and in N a strictly increasing sequence {mn}nEN such that

°

If I E Co ([0, 1]) and f > let nk be such that ~k < ~ and III - Ink 1100 < ~. Then III - E:,!1 Pi 1100 < f. The series E:=1 Pm may be viewed as a power series S ~f E:=1 anxn in which the coefficients an are in Q. For each I in Co ([0, 1]) appropriate grouping of the terms in the series S achieves the desired convergence.

[ Note 2.1.3.4: Actually the grouping is done on the groups formed by the various polynomials Pn : the series E:=1 Pn is the series in which the terms Pn can be grouped to yield the convergence of the partial sums to a given I in Co ([0, 1]).] Compare the following result, a simplified analog of the conclusion above, with the Riemann derangement theorem (Exercise 2.1.3.8. 69). Exercise 2.1.3.45. Show that there is in I. a sequence {an}nEN such that if tEl. U {oo} U { -oo} there is a grouping

(2.1.3.8)

for which the sum of the (possibly divergent) series in (2.1.3.8) is t. Newton's algorithm for finding the real root(s) of an equation of the form I(x) = 0, can fail to produce a convergent sequence of "approximants."

Section 2.1. Classical Real Analysis

95

y·axis

f(x)~ ~ (3(x - 3) - (x - 3)3 + 3)

o

x-axis

Figure 2.1.3.2. Newton's algorithm for the roots of f(x) =

o.

Example 2.1.3.9. The curve in Figure 2.1.3.2 is the graph of the equation y = f(x) and the choice of ao as the starting point for the use of Newton's algorithm leads to the real root of f(x) = O. On the other hand, the use of bo as the starting point produces a divergent sequence {bn}~=o of "approximants." 2.1.4. lR xxY

Although the study of lRx subsumes the study of lR xxY there are particularities about the latter that deserve special attention. Most of the important phenomena are illustrated in the context of lRlRxlR. Exercise 2.1.4.1. Let S be a set and let X and Y be topological spaces. Formulate appropriate definitions of uniform convergence and uniform continuity (with respect to S) for a function f in yXxs. Assume, to boot, there are uniform structures U resp. V for X resp. Y. Formulate an appropriate definition of a Cauchy net, uniform with respect to S, in yxxs. In what follows, unless the contrary is stated, the domain of each function is (a subset of) lR 2 (= lR x lR). Exercise 2.1.4.2. Show that if

f(x, y) d~f {~ OX +y

if x 2 + y2 :f 0 ifx=y=O

then f is continuous in each variable separately and yet is not continuous at (0,0) inlRx lR.

Chapter 2. Analysis

96

Exercise 2.1.4.3. Show that if !(x, y) def{~ = +y

0

t- 0

if x 2 + y2

ifx=y=O

then! has no limit at (0,0) but that for every straight line L through the origin lim(z,y) ..... (O,O), (z,y)EL !(x, y) = O. Contrast this result with that in Exercise 2.1.4.2. Convergence to (0,0) on a straight line L through the origin can be replaced by convergence on a curve C through the origin if the form of ! is appropriately modified. Exercise 2.1.4.4. Show that if if xt-O if x

and

at- 0 then! (a,e-;\)

=~

+0 as

a

--+

=0

O. If m,n E N, (m,n)

= 1,

c t- 0, and C is the curve for which the equation is y = cxIf!- then !(x, y) lim (z,y) ..... O, (z,y)EC

= O.

[ Note 2.1.4.1: Each of the three functions just described is nondifferentiable at the origin but each has first partial derivatives everywhere.] For X and Y subsets of Hausdorff spaces, for a E X \ X, for b E Y \ Y, and for a function ! in aX x Y, there are three limits to be considered at (a, b): lim

(z,y) ..... (a,b)

!(x,y)

lim lim !(x, y)

z ..... ay ..... b

lim lim !(x, y).

(2.1.4.1)

y ..... bz ..... a

Exercise 2.1.4.5. Show that for each of the functions described below exactly two of the three limits in (2.1.4.1) exist and are equal at (0,0): !(x, y)

!(

x, y

~f {z21:Y2 ~f x 2 + y2 t- 0

) ~f

o

{

Ifx=y=O

Y + x sin(!)

0

!( x,y ) ~f - {x

o

y

+ y sin(!) z

if y if y

t- 0 =0

if xt-O if x = O.

Section 2.1. Classical Real Analysis

97

Exercise 2.1.4.6. Show that for each of the functions below exactly one of the three limits in (2.1.4.1) exists at (0,0):

f( f(

x, y

ifxy::JO ) ~f{xsin(t)+YSin(~) 0 'f 0

x,y

) ~f{xi?:Y2+xsin(t) 0

1

f(x, y)

THEOREM

~f { XlJ:y2

2.1.4.1.

o

xy =

ify::JO if y = 0

+ Y sin( ~)

~f x ::J 0

If x = O.

(MOORE-OSGOOD) IF

i. X AND Y ARE SUBSETS OF HAUSDORFF SPACES AND ii. a E X \ X, b E Y \ Y j iii. limx-+af(x,y) ~f h(y) ON

iv. limy-+bf(x,y) ~f g(x)

E R,XxY j

Y\ {b}j

UNIFORMLY ON X \

THEN ALL THREE LIMITS IN

f

{a},

(2.1.4.1) EXIST AND ARE EQUAL [02].

[Remark 2.1.4.1: The result above can be generalized a bit if R, is replaced by a Hausdorff space in which there is defined a uniformity.) Exercise 2.1.4.7. Show that if

f(x, y)

~f { O:;~::

if x 2 + y2 ::J 0 ifx=y=O

then lim lim f(x, y) = 1 x-+o y-+o lim lim f(x, y) = -1 y-+Ox-+O (each repeated limit exists but the two are not equal). The importance of the conditions a E X \ X, . .. , is emphasized by Exercise 2.1.4.8 below.

Exercise 2.1.4.8. Show that if f is defined on R,2 and

f( x

,y

) ~f

{10

~f xy ::J

If xy

0

=0

Chapter 2. Analysis

98 then

~f h(y) =

if y ¥ 0 { 01 ify = 0 lim I(x, y) ~f g(x) = { 1 if x ¥ 0 y .....O 0 if x = 0 lim I(x, y)

x ..... O

and both limits above are uniform. Show also that nevertheless lim I(x,y) (x,y) ..... (O,O) does not exist. By considering I defined on ]R2\ { (x, y) the result with THEOREM 2.1.4.1. 97.

xy = O} reconcile

Exercise 2.1.4.9. Show that if

X2 sin(~) + y2 sin(t) 2 sin( ! ) x { I( X,y ) f y2 sin( _) ~f

o

then

I

y

ifxy¥O ifx¥Oandy=O if x = 0 and y ¥ 0 ifx=y=O

is differentiable everywhere:

Ix(x, y) = {

f y(X,y ) --

~x sin(~) - cos(~)

{2 0

y sin(t) - cos(t)

ifx

if x

¥0

=0

ify ¥ 0 ify = 0

but neither Ix nor Iy is continuous at (0,0). [Hint: Use the mean value theorem in the following form. There are two functions €l(h, k) and f2(h, k), defined if h2 + k2 ¥ 0, such that lim

(h,k) ..... (O,O)

€l(h, k) =

lim

(h,k) ..... (O,O)

f2(h, k) = 0,

and such that

I(h, k) - 1(0,0) = Ix(O, O)h + ly(O, O)k + €l(h, k)

Exercise 2.1.4.10. Show that if

I(x,y)

=

def {

-11; x Y x; x +11 O

if x 2 + y2 .,.. ...L 0 ifx=y=O

+ €2(h, k).)

Section 2.1. Classical Real Analysis

99

then: i.

if x ;C 0 if x = 0 if y ;C 0 ify = 0

(each first partial derivative exists everywhere); ii. each first partial derivative is continuous everywhere; iii. Izy(O,O) = 1 lyz(O,O) = -1. Exercise 2.1.4.11. Show that if L is defined according to

I : R. 2 \ L

3 (x, y)

1-+

{X02

= {(x, y)

x

~

if x > 0 and y otherwise

0, y

= O} and I

>0

then: i. Izz, Izy, Iyz, and Iyy are continuous (whence Izy ii. Iy == 0 in R.2 \ L; iii. I is not independent of y in R.2 \ L.

[ Note 2.1.4.2:

= Iyz);

Thus the implication

I y == 0 =? I

is independent of y

is not valid without regard to the nature of the domain of I. The implication is valid, e.g., if every vertical line meets the domain of I in an interval.] Exercise 2.1.4.12. Show that I(x, y) ~f (y-x 2 )(y-3x 2 ) has no local I is confined to any line L through the origin then the resulting function has a local extremum at (0,0) (cf. Example 2.1.2.3. 60). [Hint: In every neighborhood of (0, 0) there is a point (0, b) where I is positive and there is a point (a, a2 ) where I is negative. On the vertical axis I has a minimum at (0,0). On the horizontal axis I has a minimum at (0,0). On the line for which the equation is y = mx, m;c 0, extremum at (0,0). On the other hand, show that if

Chapter 2. Analysis

100

g'(O)

= 0,

g"(O)

= 2m2,

and 9 has a local minimum at (0,0).]

Exercise 2.1.4.13. Show that if

f( x,y ) ~f and x =1= 0 then

11 11

{

3 ~ ~e- v ~

f(x, y) dy

if y > 0 ify = 0

= xe- x2

fx(x, y) dy = e- x2 (1 - 2x 2 ).

Show that

and, consequently,

dd

x

11

f(x,y)dy

0

I x=o

= 1 =1=

11

af(O a' y) dy.

0

X

Exercise 2.1.4.14. Show that if y-2

f(x,y) ~f { _x- 2

o then

if 0 < x < y < 1 if 0 < y < x < 1 otherwise if 0 :5 x :5 1 and 0 :5 y :5 1

1111 1111

f(x,y)dxdy = 1 f(x,y)dydx =-1.

Exercise 2.1.4.15. Show that if 1

1

o i2 2

-~

0

... ...

)

Section 2.1. Classical Real Analysis

101

then 00

La

mn

= 2- m +1 ,

mEN

n=l 00

"~amn =- 2- n + 1 ,nEn ~ m=l 00

00

L La

mn

=2

mn

=-2.

m=l n=l 00

00

LLa n=lm=l

[ Note 2.1.4.3: In each of the last two instances there is a counterexample to a weakened version of Fubini's theorem. The condition IxxY Ig(x, y)1 dp.(x, y) < 00, for the appropriate interpretation of the product measure p., is violated.] In 1900 in Paris, Hilbert delivered a lecture on 23 open problems in mathematics [Hil]. Problem 13 posed a question of which a generalized form is: Can a continuous function f of more than one real variable be expressed as a sum of superpositions [compositions] of continuous functions of one variable? Example 2.1.4.1. If

9

( ) def Z

z

2

="4 2

h(z) ~f_~ 4

def p(Z ) Z

=

def q ( Z ) =-z

then f(x, y) ~f xy = g[P(X) - q(y)]

+ h[P(x) + q(y)].

(2.1.4.2)

Despite simple instances such as that in Example 2.1.4.1 Hilbert conjectured that the answer to his (more restricted) question was "No." In a sequence of papers [Arn, Ko3, Lor, Sp] Hilbert's generalized question was discussed, answered "Yes" by Kolmogorov, and given the following dramatic resolution by Sprecher.

Chapter 2. Analysis

102

THEOREM 2.1.4.2. ASSUME 2 ~ N E N. THERE IS A MONOTONICALLY INCREASING FUNCTION tPN : [0,1]1-+ [0,1] SUCH THAT tPN ([0, 1]) [0,1] AND tP E LIP [In 2/ In(2N + 2)]. IF 6 > THERE IS IN (0,6] A RATIONAL NUMBER f AND THERE ARE A CONTINUOUS FUNCTION X AND A CONSTANT ,\ SUCH THAT IF 2 ~ n ~ N THEN EVERY REAL FUNCTION I OF n REAL VARIABLES MAY BE EXPRESSED ACCORDING TO THE FOLLOWING EQUATION:

=

°

(2.1.4.3) THE CONTINUOUS FUNCTION X DEPENDS ON I, THE CONSTANT ,\ IS INDEPENDENT OF I, AND tPN, WHICH IS A FORTIORI CONTINUOUS, IS INDEPENDENT OF I AND n.

[Remark 2.1.4.2: The representation (2.1.4.3) is a vast improvement over (2.1.4.2) in that there is only one function X rather than two functions 9 and h and there is only one function tP N rather than two functions p and q.] The proof of THEOREM 2.1.4.2 is long and detailed and is not reproduced here. However, some of its underlying ideas and techniques, namely coverings and separating functions, are reminiscent of those in the proof of the Stone- WeierstrajJ theorem [HeSt, Loo]. The main ingredients of the argument may be described as follows.

2n + 2 ~ 'Y E N. For k in N there is a finite set Ak of indices such that Sk ~f {S2(i)hE A k is a set of pairwise disjoint cubes in ]Rn.

t. Assume

Each cube is of diameter not exceeding 'Y- ksuch that if

then for each k in N the union

1.

There is a vector v

U;,!.;n SZ covers the unit cube

m + 1 times. Furthermore the labellings are such that if io E Ak there are in Ak uniquely determined indices i o, ... ,i2n such that

n 2n

SZ(i q ) '" 0.

q=O

ii. If {hqh$q$n+m is a set of continuous functions that separate points of

en and if, for each k, hq [SZ(i)]

n hr [SkU)] = 0,

r '"

q, i '" j

(2.1.4.4)

Section 2.2. Measure Theory

103

then for every continuous function f there is a continuous function X such that f can be represented in the form m

f(Xb ... ,xn ) =

LX [hq(Xb ... ,xn )]. q=O

iii. If 0 > 0 and ko E N is such that for k ~ ko, f ~f (-y - l)-l1'- k 5 0, there is a monotonically increasing function t/J mapping £1 on itself and there is a constant A such that the functions n

gq(Xb'" ,xn ) ~f

L

Amt/J(Xm

+ fq) + q, 05 q 5

2n,

m=l

behave like the hq in (2.1.4.4). iv. If k ~ ko the sets SZ(i) are defined via the parameters 1', q, k, and i restricted to the set A% ~f [(1'k - 1)fq, 1'k + (1' k - 1)fq] n N as follows: .) def . -k

= t1'

ek ( t

o def l' k

2

= 1'-11'

-k

Ek(i) ~f [ek(i), ek(i) EZ(i) ~f [ek(i) If iq ~f {i1q, ... , i nq }

+ Ok] fq, ek(i) + Ok -

fq].

c A% the corresponding Cartesian product

II

EZ (i pq )

l:S;p:S;n

is the cube SZ (i q ) in an. v. The construction of the function t/J is based on the intervals EZ(i) much as the construction of the Cantor function Co is based on the intervals deleted in the formation of the Cantor set Co. In [Sp] all the details are given while Lorentz gives a perspicuous presentation for the case in which N = 2 [Lor]. 2.2. Measure Theory

2.2.1. Measurable and nonmeasurable sets The setting for discussion of measure theory is a measure situation (X, S, 1'), i.e., a set X, a IT-ring S consisting, by definition, of the measurable

Chapter 2. Analysis

104

subsets of X, and a countably additive set function J.I., here called a measure: J.I. : S 3 A 1-+ J.I.(A) E [0,00]. Very frequently X is an for some n in N, S is the u-ring S(K) generated by the compact sets of an or the u-ring C consisting of all Lebesgue measurable sets in an, and J.I. is n-dimensional Lebesgue measure An (AI ~f A). In a locally compact group G the u-ring S is the u-ring S(K) generated by the compact sets of G and the measure J.I. is a (tmnslation-invariant) Haar measure: A E S,x E G

"* xA E Sand J.I.(A) = J.I.(xA).

The facts about measure theory are discussed in some detail in [Habn, Loo, Rao, Roy, Rud, Sz.-N]. Important results in measure theory as it applies to Haar measure on locally compact groups, e.g., to Lebesgue measure An on an, are:

i .. a set of measure zero, i.e., a null set, contains no nonempty open set; ii. if A is a set of positive measure then AA-I contains a neighborhood of the identity; since a is a group in which the binary operation is written additively the set AA-I in a is written A - A. [Remark 2.2.1.1: Although measurable sets and measurable functions are treated in separate Subsections of this book, there is no essential distinction between them. If one accepts measurable set as a primitive notion, then a measurable function is nothing more than the limit of a sequence of simple functions, each of which is a linear combination of characteristic functions of measurable sets. If one accepts, e.g., as in the development of the Daniell integml [Loo, Rao, Roy], measumble function as a primitive notion (derived in turn from an even more elementary notion, that of a nonnegative linear functional defined on a linear lattice of extended a-valued functions), then a measurable set is nothing more than a set for which the chamcteristic function is a measurable function. Thus a result about measurable sets has its counterpart in a result about measurable functions and vice versa. Similar comments apply to sets that have, in an, n-dimensional content and to functions that are Riemann integmble over subsets ofalRR [03]. For purposes of illustration, the somewhat artificial distinctions above are useful.] THEOREM 1.1.4.2. 7 is not an accident. Indeed, Sierpinski [Si2] established the following result. THEOREM 2.2.1.1. IF B def = { x>. } >'EA IS A HAMEL BASIS FOR a OVER Q THEN B IS LEBESGUE MEASURABLE IFF A(B) = O.

Section 2.2. Measure Theory

105

PROOF. Just the "only if" requires serious attention. Assume B is a measurable Hamel basis and that >..(B) > O. It follows that B-B contains a neighborhood of 0, in particular infinitely many rational numbers. Assume that rand s are different nonzero rational numbers in B - B. Then r '" s, rs '" 0, and there is in Qat and in B elements x~p 1 ~ i ~ 4, such that

= S = tr = s = X~3 r

X~l

X~2

X~3

X~4

X~4

= t(X~l -

X~2)'

in contradiction of the linear independence of B over Q.

D

The set of Borel sets in IRn is 8(0), the a-ring generated by the open sets in IRn. In IRn the a-rings 8(F) (generated by the closed sets) and 8(K) (generated by the compact sets) are the same as the set of Borel sets. [Note 2.2.1.1: If IRn is given the discrete topology so that every set is both open and closed and a set is compact iff it is finite, then

8(0)

= 8(F) ¥8(K).J

In [Si2] there is also a proof of the next result. THEOREM 2.2.1.2. No HAMEL BASIS B CAN BE BOREL MEASURABLE. [Remark 2.2.1.2: Hence the Hamel basis B of THEOREM 1.1.4.2. 7 is a non-Borel subset of the Borel set Co.

The cardinality of the set of all Borel sets is #(IR) whereas the cardinality of 1'(Co), the power set of Co, is 2#(IR). It follows, without reference to THEOREM 2.2.1.2, that there are non-Borel sets of measure zero. Since, for any function !, Discont(f) is an Fu it follows that there are null sets that cannot be Discont(f) for any function !.J THEOREM 2.2.1.3. IN EVERY NON EMPTY NEIGHBORHOOD U OF 0 IN IR THERE IS A HAMEL BASIS FOR IR OVER Q. PROOF. Let r be a positive rational number such that (-r, r) C U and let H be some Hamel basis for IR over Q. For each h in H there is in Z a unique mh such that mhr ~ h < (mh + 1) r. Let kh be h (mh + 1) r. Then kh E (-r, r). If K is a maximal linearly independent subset of {kh : h E H } U {~} then K is a Hamel basis for IR over Q and K C (-r,r) C U.

D

Chapter 2. Analysis

106

The result above is a special case of a more general phenomenon: In any neighborhood U of the identity in a Lie group G there is a relatively free subset [Ge5]. In the Lie group lR a maximal relatively free subset of U is perforce a Hamel basis. The existence of nonmeasurable (Lebesgue) subsets of lR cannot be based on a cardinality argument. The Cantor set Co has the cardinality of lR. Since A(Co) = every subset of Co is Lebesgue measurable it follows that the cardinality of the set £, of all Lebesgue measurable sets is 2#(1R) , which is also the cardinality of the power 8et P(lR) ~f 21R of lR.

°

Example 2.2.1.1. The map (J : lR 3 t f-+ e2...it algebraically and measure-theoretically identifies lR/Z with the compact mUltiplicative group 'Jl', and identifies Q/Z with a countable and infinite subgroup H of 'Jl'. The Axiom of Choice implies that there is in 'Jl' a set S consisting of exactly one element from each of the cosets of H. If r1 and r2 are different elements of Hand r 1S = r 2S then in S there are 81 and 82 such that r1S1 = r282. But then 81 and 82 are in the same coset of H, whence the nature of S implies 81 = 82, i.e., r1 = r2, a contradiction. Thus

If S is measurable then, since A transferred to 'Jl' is again "translation" invariant, S and all the rS have same measure a: A(rS) == a. Then

EITHER a = 0, in which case A('Jl')

=L

A(rS)

= 0,

A(rS)

= 00.

rEH

OR a

> 0, in which case A('Jl')

=L rEH

Since neither conclusion is correct, S is not measurable nor is its counterpart (J-1(S) n [0, 1). Example 2.2.1.2. Any countable and infinite subgroup G of'Jl' may serve instead of H in the discussion above. In particular, if a is an irrational def 2 . real number and = e ... ,0< then either of the subgroups

e

Section 2.2. Measure Theory

107

may be used. Note that: i. B is a subgroup of index 2 in Aj ii. BneB = 0 and A = Bl:JeBj iii. because a is irrational both subgroups A and Bare (countably) infinite

dense subgroups of the compact group T. Let P consist of exactly one element of each coset of A and let M be PB. If MM- I neB -:F 0, i.e., if

and XIX2"l E eB, then PIP2"l E eB C A and so, owing to the nature of P, PI = P2. Thus XIX2"l = bI b2"l E B, i.e., xIx2"l E eB n B = 0, a contradiction whence M M-I neB = 0. If L is a measurable subset of M and A(L) > 0 then MM- I :::> LL-1, which contains a T-neighborhood of 1 (cf. THEOREM 1.1.4.1. 5) and thus an element of the dense set eB, a contradiction. It follows that the inner measure of M is zero: A*(M) = o. For x in T there is in Pap such that xp- I ~f a E A. If x f. M then a f. B whence for some b in B, x = peb E peB = eM. Thus

and so A* (MC) = o. The inner measure A* and outer measure A* are set functions such that for each measurable set P, A*(P n M)

+ A* (P n M = A(P), C)

whence A*(P n M) = A(P), in particular, A*(M) = 1 > 0 = A*(M). The set (J-I(M) ~f Min JR has properties analogous to those of M. i. The set M is nonmeasurable, A* (M)

= 0,

and A* (M)

= 00.

ii. The set M is thick and for every measurable subset P of JR, A*(P n M)

= 0 while A*(P n M) = A(P).

Exercise 2.2.1.1. Let G be an infinite subgroup of T. Show: i. the identity is a limit point of Gj ii. every infinite subgroup of T is dense in T. iii. the compact subset 1 x T is a nowhere dense infinite subgroup of the compact group T2.

The Cantor set Co is one of a family of nowhere dense perfect sets. The construction of a typical member of the family is a modification of the construction of the Cantor set Co.

Chapter 2. Analysis

108

Example 2.2.1.3. If € E Q n (0, 1) let an be €. 2- 2n +l, n E N. Then 2n - 1 a n = €. Let 7"1 and 7"2 be two transcendental numbers such that 7"1 < 0, 7"2 > 1 and 7"2 - 7"1 = 1 + 2€ and let {TkhEN be an enumeration of the set S ~f [7"1> 7"21 n A consisting of the algebraic numbers in [7"1> 7"21 ~f I. Let {Imn, mEN, 1:::; n :::; 2m- I } be the set of open intervals deleted from [0,11 in the construction of the Cantor set Co. The first open interval III of length 3- 1 , the next two 121> 122 each of length 3- 2 , ••• are, for the current construction, replaced by open intervals J1> J 2, J3, . .. so that the endpoints of each I n are transcendental and: E:'=1

A(Jl) :::; al A(Jk):::; a2,

k = 2,3

Furthermore let J 1 be placed to contain Tl: Tl E J 1 . Let Tkl be the first Tk not in J 1 • There is a first n, say n1> such that J n1 may be chosen to contain Tkl and to be disjoint from J 1 : Tkl E J nll Jl n J n1 = 0. By induction one may find a sequence {Tkp}PEN in S and a sequence {Jnp}PEN in {In}nEN such that i.

Tk p

is the first

Tk

not in

ii. Tk p E Jnp ;

In.

It follows that: iv. J ~f J 1 l:J (l:JPENJnp ) is an open subset of I;

v.

1\ J ~f

CJ contains no algebraic numbers, i.e., consists entirely of transcendental numbers; vi. CJ is nowhere dense in [7"1,7"21 and perfect; vii.1+2€~A(CJ)~1+€.

Exercise 2.2.1.2. Show that CJ is nowhere dense in

[7"1> 7"21

and per-

fect. [Hint: The complement I \ C J of C J in I is open and is dense in I. To show C J is perfect it suffices to prove that each of the count ably many endpoints of the intervals J np is a limit point of CJ·1

Section 2.2. Measure Theory

109

Exercise 2.2.1.3. Repeat the construction in Example 2.2.1.3 with the following modification: 0 < Tl < T2 < 1, T2 - Tl > 1 - f. The resulting set, say D, should consist entirely of transcendental numbers, be nowhere dense in [TbT2]' be perfect, and have measure greater than (T2 - Tl) - 2f. Construct a sequence {Dn }nEN of sets so that each consists entirely of transcendental numbers, is nowhere dense in [0, I], and is perfect. Furthermore, the following should obtain:

Dn C Dn+l C [0, I], n E N, A

(U

Dn)

~f A(Doo) = 1.

nEN

Hence the set Doo: i. consists of transcendental numbers; u. is dense in [0, I]; iii. is an F.,.; iv. is a set of the first category in [0, I];

Furthermore E ~f [0, I] \ Doo is a null set of the second category. Exercise 2.2.1.4. Assume the endpoints of the closed interval [a, b] are rational. In [a, b] construct a Cantor-like set Ca,b such that A(Ca,b) = O. Show that the union H~f

Ua
a,bEQ,

is a null set and that every x in R. is a point of condensation of H (cf. Exercise 2.1.1.18. 52). [Remark 2.2.1.3: Although every point x of R. is a point of condensation of the set Kof irrational numbers, Kis quite the opposite of a null set since its complement Q is a null set.] When G is a group and A and B are subsets of G the sets AB and AB- 1 frequently enter arguments about G. When G is abelian and its binary operation is written as + the corresponding sets are the sum set A + B and difference set A - B. When there is a measure situation (G,S,IL), in particular when G is a locally compact group and IL is translation invariant (Haar) measure, there arises the question: If A and B are measurable is AB measurable? A negative answer is provided by the next results. Example 2.2.1.4. Let G be R. 2 and let E be a nonmeasurable subset of R.+. If A ~f ({O} x E) U (E x {O}) then A2(A) = 0 whence A is measurable (A2). Since

A + A = [{O} x (E

+ E)]l:J [(E + E)

x {0}]l:J [E x E] ~f Pl:JQl:JR

Chapter 2. Analysis

110

and since ~2(P) = ~2(Q) = 0, if A + A is measurable then so is E x E. However if x E E the section (E x E)z is E and so not almost every section of A + A is measurable, a contradiction of Fubini's theorem. Example 2.2.1.5. Let B be a Hamel basis contained in the Cantor set Co. If 8 1 ~f UrEQ(rB) then ~(81) = 0 and 8 1 = -81. Let 8n +l be 8n + 8n (= 8n - 8 n ), n E N. If x

~

L abb E 8

n , ab E

Q

bEB

then x is a sum of not more than 2n - 1 members of 8 1 and hence at most 2n - 1 of the coefficients ab are not zero. Since Co + Co = [0,2) it follows that UnEN 8n = IR and if each 8n is measurable then one of them, say 8no ' has positive Lebesgue measure: ~(8no) > O. Hence for some M in N

contains a neighborhood U ~ (_2- M , 2- M ). If KEN and 2n o


Q \ {O} :::> {ri! ... , rK} (a K-element set) K sup Irkl < TM 1:::;k:::;K

then whenever {bi! ... , bK} is a K-element subset of B it follows that K

L rkbk E U \ 8no

+l

=

0,

k=1

a contradiction. Thus some 8 n is nonmeasurable and if 8 n1 is the first nonmeasurable 8 n then n1 > 1. Hence 8 n1 -1 is measurable and

is nonmeasurable. (This result was communicated to the writers by Harvey Diamond and Gregory Gelles.) Example 2.2.1.6. The Cantor-like sets Do and Dn in Example 2.1.1.3.51 and Exercise 2.2.1.3. 109 may be chosen so that ~ (Do) = 00, ~ (Dn) = On, and 00 + LnEN On = 1. In that event Do and each Dn is nowhere dense whence the corresponding set D that is the union of them all is of the first category and ~(D) = 1. The complement [0,1) \ D is perforce

Section 2.2. Measure Theory

111

of the second category and its measure is zero. Relative to [0, 1] the set D is thick. Exercise 2.2.1.5. Let A be the (countable) set of endpoints of the intervals deleted in the construction of Co and let B be Co \ A. Show that A and B are disjoint nowhere dense sets such that each point of A resp. B is a limit point of B resp. A. From Example 2.2.1.6 it follows that category and measure are, at best, loosely related. There are sets of the first category that have measure zero, e.g., Q, and sets of the first category that are thick, e.g., on each interval tn, n + 1], n E Z construct a set Pn just like D. Then set P ~ UnEZ Pn is such that A (JR \ P) = O. There are sets of the second category, e.g., JR \ P, that have measure zero and sets of the second category, e.g., lR, that are thick [Ox]. Category is not preserved under homeomorphism. To see this call a linearly ordered set A ordinally dense if it has neither first nor last member and between any two members there is a third. For example Q in its natural order is ordinally dense; Z is not ordinally dense in its natural order; neither N in its natural order nor any other well-ordered set in the order of its wellordering is ordinally dense. Two ordered sets are ordinally similar if there is an order-preserving bijection between them. THEOREM 2.2.1.4. IF A def{} an nEN AND B def{} bn nEN ARE TWO (COUNTABLE) ORDINALLY DENSE SETS THEY ARE ORDINALLY SIMILAR.

=

=

PROOF. The order-preserving bijection is defined by induction: Let a1 be a1 and 131 be b1. If all" ., a n -1 and 1311.' ., f3n-1 have been chosen so that ai +-+ f3i, 1:5 i :5 n - 1, is an order-preserving bijection and n is even let an be the first am not yet chosen and let f3n be the first bm not yet chosen and order-related to {f311"" f3n-d as an is order-related to {all"" an-d. If n is odd let f3n be the first bm not chosen and let an be the first am not chosen and order-related to {all' .. , an-d as f3n is order-related to {f311' .. ,f3n-d. The method of choice is such that {an }nEN = A and {f3n }nEN = B and the bijection an +-+ f3n is order-preserving.

o In particular, the set {In}nEN of intervals deleted in the construction of Co is ordinally dense if I -< I' is taken to mean that I is to the left of I'. Let the set (0,1) n Q ~f {rn}nEN ~f A, which is also ordinally dense, be in bijective order-preserving correspondence with {In}nEN. Define f on UnEN In so that if x E In then f(x) = Tn. Thus f is monotone increasing, its range {Tn}nEN is dense in [0,1], and so f may be extended to a continuous

Chapter 2. Analysis

112

monotone increasing function, again called /, on [0, 1]. Let B be Co shorn of the endpoints of the deleted intervals. Then / maps [0,1] \ B onto {rn}nEN. Owing to the ordinal similarity of {rn}nEN and {In}nEN, / is increasing on [0,1], strictly increasing and bicontinuous on B, and also /(B) = [0,1] \ Q. Thus Co (and hence any Cantor-like set) shorn of its endpoints is homeomorphic to the set of 1[0,1] of irrational numbers in [0,1]. However B, as a subset of Co, is nowhere dense and hence of the first category while 1[0,1] is of the second category. Exercise 2.2.1.6. Show that B above is homeomorphic to R. \ Q. Hence a nowhere dense set B is homeomorphic to a dense set R. \ Q. More interesting phenomena in the relationships between measuretheoretic and topological properties arise in the context described below. The Cantor set Co, {O,I}N, the countable Cartesian product of the two-point set {O, I} in its discrete topology, is the source of some of these phenomena. More generally, for an arbitrary infinite set M, let {O, I}M be the (possibly uncountable) Cartesian product or dyadic space 1)M. The weight W of a topological space X is the least of the cardinal numbers W such that the topology of X has a base of cardinality W. If U~f {Uj};EJ is a base and #(J) = W then U is a minimal base for X. LEMMA

2.2.1.1.

i. EVERY SEPARABLE METRIC SPACE IS THE CONTINUOUS IMAGE OF A SUBSET OF THE CANTOR SET.

ii. EVERY COMPACT METRIC SPACE IS THE CONTINUOUS IMAGE OF THE CANTOR SET.

if

EVERY COMPACT

Hausdorff SPACE X OF WEIGHT

# (M)

IS THE CON-

TINUOUS IMAGE OF A CLOSED SUBSET OF 1)M.

iii. EVERY COMPACT totally disconnected METRIC SPACE IS THE HOMEOMORPHIC IMAGE OF A SUBSET OF THE CANTOR SET.

iv. EVERY COMPACT TOTALLY DISCONNECTED perfect METRIC SPACE IS THE HOMEOMORPHIC IMAGE OF THE CANTOR SET.

[The fundamental idea behind the proof of if can be described as follows. If #(J) = # (M) and if

Edef{} = ej jEJ E 1)M let U ~ {Uj };EJ be a minimal base for X. For each j in J there is defined a dyad of closed sets:

Ai. ~f 3

{

Uj X \ Uj

if i if i

°

= = 1.

Section 2.2. Measure Theory Then A~ ~f

{e : A~

=I-

njEJ

A;'

113

is either 0 or a single point. Let:3 be

0 }. Then:3 is a closed subset of V M, the map F::33eI-+A~

is continuous, and F(:3) = X. (If #M = #(N) the map F can be extended to a continuous map ~ : V M 1-+ X, i.e., ii.)] See [AH, Bou, Cs, Eng, HeSt, Kur, Rin] for detailed proofs of the various parts of LEMMA 2.2.1.1. Note that if is an imperfect counterpart of ii. In fact, there is no perfect counterpart to ii, as the contents of Exercises 2.2.1. 7 and 2.2.1.8 below show [Eng]. Exercise 2.2.1.7. Let X be a set such that #(X) > #(N). Fix a point Xo in X. Define a topology by declaring that a subset A of X is open iff Xo f/. A or X \ A is finite. Show that: i. X is a compact Hausdorff space; n. everyone-point set {x} other than {xo} is open; iii. the weight of X is #(X).

Assume that for some M there is a continuous surjection f Then, since each point other than Xo in X is open, the set

: V M 1-+ X.

consists of uncountably many pairwise disjoint open subsets of V M . Exercise 2.2.1.8. Let U be a set of basic neighborhoods for V M . Show that if the elements of U are pairwise disjoint then U is empty, finite, or countable. Show that if 0 is a set of pairwise disjoint open subsets of V M then 0 is empty, finite, or countable. Why do the preceding conclusions show that X in Exercise 2.2.1.7 is not the continuous image of some dyadic space V M ? An arc-image resp. open arc-image '1* is, for some arc resp. open arc '1 in C ([0,1], X) resp. C «0, 1), X) the set '1 ([0,1]) resp. '1 ( (0, 1)). If '1 is injective, the image is simple. The endpoints of an arc-image '1* are '1(0) and '1(1). If '1(0) = '1(1), 'Y*is closed; if, to boot, '1 is injective on (0,1), '1* is a simple closed curve-image or Jordan curve-image. [ Note 2.2.1.2: The image '1*, a subset of a topological space X, is by definition different from '1 itself, which is a continuous function. (Nevertheless, by abuse of language, the distinction is occasionally blurred and, e.g., "A Jordan curve in ]R2 separates the plane," is an acceptable substitute for the more accurate, "A Jordan curve-image in ]R2 separates the plane.")]

Chapter 2. Analysis

114

For an arc "(: [0,1]1-+ (X1(t), ... ,xn(t)) E IRn, the length l("() of"( is defined to be n

L (Xj (ti) -

Xj (ti_1))2, N E N.

j=1

However, the length L ("(*) of the arc-image "(* is the infimum, taken over the set P of all parametric descriptions s of "(* , of l ("( 0 s). Each parametric description is a continuous autojection s : [0,1]1-+ [0,1]. Thus L ("(*) ~f

inf

{B : BEP}

l("( 0 s).

The length of an arc and the length of the corresponding arc-image can be quite different. The length l("() can be infinite while L ("(*) is, in the usual geometric sense, finite. Example 2.2.1.7. z. Let "( be {

X

Y

= cos47rt = sin47rt

' t E [0,1].

Then l("() = 47r, whereas "(* is a circle of radius 1 and L ("(*) = 27r. ii. Let g be a continuous nowhere differentiable function on [0,1] and let "( be [0 1] 3 t 1-+ = g(t) , y = g(t).

{X

Then, since g is not of bounded variation on any nondegenerate interval, if ~ a < b ~ 1 the arc defined by restricting "( to [a, b] is nonrectifiable: l("() = 00. On the other hand, the arc-image "(* is a straight line segment and

°

L ("(*)

= V2 ( tE[O,1j sup g(t) -

inf g(t))

tE[O,1)

which, owing to the continuity of g, is finite. Example 2.2.1.8. The unit n-cube or parallelotope r (the topological product of n copies of [0,1]) is the continuous image of Co. The map t : Co 1-+ pn may be extended linearly on the closure of each interval deleted in the construction of Co and the image of the resulting map T is an arc-image T ([0,1]) that fills r. Since Co is totally disconnected whereas pn is connected neither t nor T is bijective.

Section 2.2. Measure Theory

115

When n = 2 the continuous map t transforms a set of one-dimensional measure zero onto a set of two-dimensional measure one. Let 11'1 be the projection of R,2 onto its first factor: 11'1 : R,2 3 (x, y) 1-+ x. If A is a nonmeasurable subset of [0, I] x {O} then D ~f r1(A), as a subset of the null set Co, is (Lebesgue) measurable whereas 11'1 0 t(D) (= A) is a nonmeasurable subset of [0, I]. Since T is an extension of t it follows that U ~f 11'10 T is a continuous map of [0, I] into itself and U maps a null set of [0, I] onto a nonmeasurable set.

°

Example 2.2.1.9. Assume ~ 0: < 1. In each factor of p2 construct a Cantor set COt so that A(COt ) = 0:. Then the topological product of the two sets COt is a compact set C~2 such that A2(C~2) = 0: 2 • Each COt is the intersection of a decreasing sequence {K; hEN of compact sets and each K; is a finite union of disjoint closed intervals all of the same length: Kl. the complement of the first open interval deleted in the construction of COt, consists of 21 disjoint closed intervals, I u , lt2' arranged in natural order from left to right in [0, I]. Similarly K;, the complement of the union of the first 2; - 1 open intervals deleted in the construction of COt, consists of 2; disjoint closed intervals, 1;1, ... ,I;2j arranged in natural order from left to right in [0, I]. The construction proceeds in sequence of stages of associations between intervals Imn and their Cartesian products Imn x Im'n" At stage 1 associate the 2 2X1 = 4 1 intervals 121 , ... ,122 2 of K2 with 4 1 sets in [0, I] x [0, I] as follows:

Having completed stages 1, ... ,j - 1, at stage j: 2'

.

.

i. associate the 2 ' = 4' intervals 12;,1, ... ,I2;,22j of K 2; with the 4' sets I;p x I;q, 1 ~ p,q ~ 2; in [0, I] x [0, I]; ii. map each interval deleted from [0, I] on to a line segment connecting two adjacent components of K 2; x K 2;.

In Figure 2.2.1.1 there is an indication of the associations made in the first two performances of the procedure just described. Subsequent associations are made similarly, by inbreeding, i.e., by repeating in each subinterval and correspondingly in each subsquare the construction just employed in the original square, and by continuing the repetition process endlessly. Although the construction is repeated in each stage, the orientation of the constructions in the subsquares must be such as to permit the connections indicated in Figure 2.2.1.1. Let IC; be the compact connected set consisting of K 2; x K 2; together with the line segments connecting its components (cf. ii above). Then

Chapter 2. Analysis

116 ICj+1

A2

(IC) ~

1

n

jEN IC j ~f IC is the homeomorphic image of [0,1]: IC = IC is a simple arc-image and since C~2 C IC it follows that

C IC j and

e ([0, 1]), i.e., 0: 2 •

2

3

4

5

121 XI 22

122 X 122

3

5

6

4

2

6 121 X/ 21

1

122 XI 21

7

Figure 2.2.1.1. The first steps of the repetition/inbreeding process.

Section 2.2. Measure Theory

117

Exercise 2.2.1.9. Show that e is a homeomorphism. [Hint: The map e is bijective on each of the intervals deleted in the construction of Co.. The set of those intervals is dense in [0,1]. Hence if x < y :::; 1 and one of x and y is not in one of those intervals, then (possibly another) one of those intervals is a proper subset of [x, y]. Hence, for some jo in N, e(x) and e(y) are in different components of Kjo x K jo ' in particular, e(x) =f e(y). The continuity ofe follows because a) the diameters of the components of K j x K j converge to zero as j -+ 00 and b) e is linear on each of the intervals deleted in the construction of Co..]

°: :;

[ Note 2.2.1.3: The simple arc-image IC is not rectifiable. The very definition of arc-length shows that the arc-image of a rectifiable arc can be covered by rectangles forming a set of arbitrarily small two-dimensional Lebesgue measure. A similar argument leads to the following conclusion: For each n in N and each I: in (0,1] there is in pn a (nonrectifiable) simple arc-image IC such that An (IC) ~ 1 - 1:.] Example 2.2.1.10. When n = 2 the simple arc-image IC described above lies in the unit square [0,1] x [0,1] and the endpoints of IC are (0,0) and (1,0). The union of IC and the simple arc-image

B ~f ({o} x [0, -6]) U ([0, 1] x {-6}) U ({I} x [0, -6]) is a Jordan curve-image C that is the boundary of a region R. Since

it follows that

Hence A2(R) < A2 (IC) if 6 < 1 - 21:, in which case the measure of R is less than the measure of the Jordan curve-image C that bounds R. In fact, for a positive I: there are a Jordan curve C and the region R bounded by C so that

Exercise 2.2.1.10. Show that a compact convex set in a separable topological vector space is an arc-image.

118

Chapter 2. Analysis [Hint: A separable topological group is metrizable [Kakl].]

There are nowhere dense ("thin") sets of positive (Lebesgue) measure, e.g., Cantor-like sets of positive measure. Besicovitch [Bes2] used such sets to construct in 1R3 a homeomorphic image BES (for Besicovitch) of the surface 8 1 of the unit ball

B1~f{(X,y,Z): x,y,zEIR,X2+y2+Z2~1} so that A3(BES) is large while the surface area A (BES) of BES is small. If 11, A > 0 there is in 1R3 a surface BES, homeomorphic to 8 1 and such that A3 (BES) > A while the surface area A (BES) of BES is less than 11. Proceeding by analogy with the definition of arc-length for a curve, one is led to suggest that the area of a surface 8 in 1R3 be defined as the supremum of the set of areas of the polyhedra inscribed in the 8. However phenomena such as that in Exercise 2.2.1.12. 123 below suggest the inadequacy of so simple an approach. The construction originated by Besicovitch and described below dramatizes even further the need to reformulate a proper theory of surface area. For example, some proper definition of surface area is necessary if there is to be a satisfactory statement, not to mention a satisfactory resolution, of the famous problem of Plateau. For a given Jordan curve-image C in 1R3 find in 1R3 a surface 8 bounded by C and of least surface area.

Example 2.2.1.11. Assume 4M 3 > A, 0 < a < 1, and 11 > 210 > O. Let K denote the cube [- M, M]3 in 1R3. The cube K is subjected to two operations performed in succession and then repeated endlessly. i. Shrinkage by a: replace K by Ko ~f [-aM, aM]3, 0

< a < 1, situated co centrally inside K and with its faces parallel to those of K j ii. Subdivision: by passing bisecting planes parallel to the faces of Ko divide it into eight congruent subcubes: K!, ... , K~. Inbreed, i.e., repeat the operations i,ii above on each of the eight subcubes, on each of the 8 2 subsubcubes, ... , on each of the 8n subsub ... subcubes.

Exercise 2.2.1.11. Let the intersection of the set of all cubes, subcubes, subsubcubes, ... be D. Then D is a dyadic space, a three-dimensional analog and homeomorphic image of the Cantor set. Calculate the measure of D in terms of M and a and thereby show that for some a the three-dimensional measure of D can be made arbitrarily close to but less than 8M3 , the volume of the original cube.

Section 2.2. Measure Theory

119

Figure 2.2.1.2. The Besicovitch construction. Only two of the first eight "ducts" are shown.

The next goal is to construct a polyhedron II containing (infinitely) many faces and edges and such that among the vertices of II are all the points of D. The procedure given next provides such a polyhedron. As a polyhedron II consists of polygonal faces and thus has a well-defined surface area. The polyhedron constructed below has small area.

120

Chapter 2. Analysis

On one face of K construct a square 8 of area not exceeding i. Note that 8 is homeomorphic to a hemisphere. The idea is to distort 8 in a thorough and systematic manner so that 8 is formed into a polyhedron of the kind described above. From 8 excise eight disjoint pairwise congruent subsquares each of area 61 not exceeding 3£2 and more narrowly delimited below (cf. Figure 2.2.1.2 above). On one face of each of the first eight subcubes construct a square congruent to one of the eight subsquares excised from 8. Again by inbreeding, repeat this construction on each of the subsubcubes, ... , so that on one face of each subsub ... subcube there is a square from which eight congruent subsquares have been excised. In K \ Ka run eight tubes, one from each of the eight excised subsquares of 8 to one of the eight squares on the eight subcubes of Ka. The connected surfaces of the tubes are to be unions of nonoverlapping closed rectangles. The cross-sections of the tubes are rectangles - in short, the tubes are models of heating/air-conditioning ducts. The planar surface area of each tube is proportional to the perimeter of the (rectangular) cross-section. Hence, by a suitable choice of 61 , the total (planar) surface area of the eight tubes can, be brought below ~. The union of 8 1 , the surfaces of the eight tubes, and the eight squares on the surfaces of the eight subcubes is homeomorphic to 8 and hence to the surface of a hemisphere. The process just described is repeated in each of the first eight subcubes, except that a new 62 is chosen so that the total surface area of the 64 new tubes does not exceed ~, .... The basic construction (simplified) is shown in Figure 2.2.1.2. The end-product of the infinite set of tube constructions is a Medusalike set HEMIBES (hemi+BES) that is homeomorphic to the surface of a hemisphere. As one moves through a first-stage tube, then through one of the second-stage tubes emanating from it, ... , at the "other end" one arrives at precisely one point of D and each point of D is the "other end" of such a trail. Thus D, a dyadic space of positive three-dimensional measure, lies on the surface of a HEMIBES, which is homeomorphic to the twodimensional surface of a hemisphere. The total surface area of the tubes so traversed is not more than i and so the surface area of HEMIBES does not exceed f whereas HEMIBES contains D and thus the three-dimensional measure of HEMIBES can be made arbitrarily close to 8M 3 • If two "hemispheres" like HEMIBES are conjoined at their "equators" (the perimeters of their squares 8) the result BES is homeomorphic to the surface of the ball B 1 • The union of BES and the bounded component of its complement is a set B that is homeomorphic to B 1 • The area of the surface of B, i.e., the area of BES, is less than T/ whereas the three-dimensional measure of B exceeds A.

Section 2.2. Measure Theory

121

[Remark 2.2.1.4: Let C be a rectifiable Jordan curve-image in If R is the bounded component of ]R2 \ C and if l( C) = 1, then

]R2.

(The second inequality is the famous isoperimetric inequality studied in the calculus of variations.) The corresponding theorem for ]R3 should read: Let E be a homeomorphic image of 8 1 in ]R3 and assume that the surface area of E is 1: A(E) = 1. If V is the bounded component of ]R3 \ E then A3(E)

= 0 and A3(V U E) :5

1

6..fi.

Whereas (*) is true, owing to BES, (**) is false. The reader is urged to formulate other contrasts stemming from BES.] [ Note 2.2.1.4: The surface BES of B can be described parametrically by three equations: x=/(u,v), y=g(u,v), z=h(u,v), O:5u,v:51.

Since the surface of B is, for the most part planar, the functions I, g, h are, off a set of two-dimensional measure zero, linear, in particular continuously differentiable a.e. The example BES illuminates not only the problem of Plateau but also the question of defining the notion of surface. For example, the parametric description of BES in terms of I, g, h above is qualitatively indistinguishable from that of the surface of a cube or the surface of a cube to which "spines" (closed intervals) or "wings" (closed triangles) have been attached. In another direction, the ball B impinges on the circle of ideas under the rubric of Stokes's theorem, which is a vast generalization of the FTC. Stokes's theorem and, in particular the FTC, may be written in terms of the symbol interpreted as a special differentiation operator when a is applied to a (vector-valued) function and as the boundary operator when a is applied to a subset of ]Rn:

a

Stokes's theorem:

{

JaR

1= { al.

JR

(The differential notation in the equation above is omitted deliberately. The integrals are to be interpreted as formed with respect to

Chapter 2. Analysis

122

appropriate measures on 8R resp. R.) For example, if R ~f [a, bj and IE C 1 ([a,bj,a) then 8R = {a,b}, 81 = /" and the FTC reads:

f

I

J8[a,b) Similarly in

a3

~f I(b) -

I(a) = fb I' dx Ja

d~f

f

81.

J[a,b)

for a ball

Br ~f

{

x 2 + y2

(x, y, z)

+ z2

~

r2 } ,

its boundary 8Br def = Sr def =

{ (

x,y,z )

a vector-valued function

F(x, y, z) ~f (f(x, y, z), g(x, y, z), h(x, y, z)), and

8F ~f" - v· F d,.!!.f - fz

+ gy + h z ~f - d·IV F ,

the (Gaufi) version of Stokes's theorem reads in terms of the (vector) differential dA of surface area and the (scalar) differential of volume dV:

f

F

J8Br

~f f

JSr

(f(x, y, z), g(x, y, z), h(x, y, z)) . dA

=f

(fz(x, y, z)

d~f

divFdV

JBr

f JBr

+ gy(x, y, z) + hz(x, y, z))

~f

f

dV

8F.

JBr

For a smooth F, the theorem fails for the ball B and its boundary BES. Similarly, for a surface S in a3 and bounded by a rectifiable closed curve C: 8S = C, ds representing the (vector) differential of curve length, there is the formula traditionally named for Stokes:

Is

8F =

~f

Is

curlF

~f

Is (hy - gz, Iz - hz, gz - Iy) . dA

f (f,g,h).ds~f f

Jc

F.

J8S

One more comment deserves inclusion. The notion of Hausdorff dimension pP, pEa, 0 < p < 00, defined for all subsets of a metric space X, is intimately related to Lebesgue measure when X = an. For BES, 0 < p2(BES) < 00, whence

Section 2.2. Measure Theory

123

°

PP(BES) = 0, 2 < p < 00, and < p3(8) < 00, whence pQ(8) = 0, 3 < q < 00, a result more in harmony with geometric intuition [Ge7].] The length of 1 : [0,1] 1-+ JRn is the supremum of the lengths of the polygons inscribed in the curve: n

i ("'f) ~f

sup

L h(ti) - 1(ti-dll, n E N.

O=to< .. ·
Much before Besicovitch produced the construction described above, H. A. Schwarz showed that the corresponding notion of defining the area of a surface by the areas of approximating inscribed polyhedra is useless. Exercise 2.2.1.12. Let a truncated right circular cylinder Z be of radius r and of height h. For 3 ~ m, n < 00, m, n E N, draw m equally spaced circles on the surface of Z so that their planes are parallel to the base of Z. Inscribe in each circle an n-gon so that the vertices of one n-gon are positioned under or Over the midpoints the sides of an adjacent n-gon. Each vertex of a top or bottom n-gon has two neighboring vertices on an adjacent n-gon. Each vertex of every other n-gon has four neighboring vertices on two adjacent n-gons. The sides of the n-gons and the line segments connecting neighboring vertices constitute the set of edges of a polyhedron IImn with vertices lying on the surface of Z: IImn is a polyhedron inscribed in the lateral surface of Z. Each face of IImn is a triangle. Show that the area of each triangle in IImn is

Find the area Amn of IImn and show that lim inf Amn

m,n-.oo

= 27rrh,

lim sup Amn m,n-+oo

= 00

whence limm,n-+oo Amn does not exist. Show also that if 27rrh there is a function m(n) such that limn-+ oo Am(n),n = A. Related to the discussions above are the following results.

~

A

<

00

Example 2.2.1.12. In JR2 the bounded component R of the complement of a Jordan curve-image J is necessarily an open set of positive finite measure. If J is nonrectifiable it follows that for the compact set K ~f J U R, A2(K) < 00 and L(8K) = 00 (= L(R)). For example, if a) B is the set in Example 2.2.1.10. 117, b) 1 in C ([0, 1], JR) is nowhere differentiable, 1(0) = 1(1) = 0, and c) 1 is 1 : [0,1] 1-+ (t,/(t)) E JR2, then the Jordan curve-image J ~f C ~f 1* U B has infinite length. Because 1*

124

Chapter 2. Analysis

in R? is the graph of a continuous function, A2 b*) = 0 whence A2(C) = O. Let R be the bounded component of ]R2 \ J, let n be R U C, and let V be n x [0,1], a compact (measurable) subset of ]R3. Then A3(V) is finite, VO = R x (0, 1), and A3 (VO) is positive (and finite). Owing to the nonrectifiability of the curve C, the area of the boundary V \ VO of V is necessarily infinite. The set VO may be viewed as the inside of a container that can be filled with paint but the (inside) surface V \ VO of the container cannot be painted. Working in a different part of the domain of "sizes" of figures in twoand three-dimensional Euclidean space, Besicovitch [Bes1, Bes3] proved THEOREMS 2.2.1.5 and 2.2.1.6 below. The former provided a surprising solution to a problem posed by Kakeya. THEOREM 2.2.1.5. LET K:. BE THE SET OF INTERIORS K DETERMINED BY PLANE CLOSED POLYGONS (JORDAN CURVE-IMAGES) P AND CONTAINING AT LEAST ONE UNIT LINE SEGMENT (OF LENGTH ONE). IF f > 0 THERE IS IN K:. A KE SUCH THAT THE AREA OF KE DOES NOT EXCEED f AND THERE IS IN KE A DIRECTED UNIT LINE SEGMENT S THAT CAN BE MOVED CONTINUOUSLY IN KE AND, IN THE COURSE OF ITS MOTION, S POINTS IN ALL DIRECTIONS: "A UNIT LINE SEGMENT CAN BE ROTATED THROUGH 360° WITHIN AN ARBITRARILY SMALL POLYGONAL AREA." PROOF. Since K:. contains the interiors of all squares of side 2, as in Figure 2.2.1.3 below, K:. is not empty. In what follows the basic ideas go back to Besicovitch and to a device due to J. Pal [Bes 1, Bes3]. In a square of side 2 the diagonals form four isosceles right triangles T 1 , ••• , T4 having a common vertex at the center of the square (see Figure 2.2.1.3). For p ~ 3 let each side of the square be divided into n 1;£ 2P- 2 equal subintervals, each the base of a triangle having its third vertex at the center of the square. Assume translations parallel to the base of T1 can bring its n subtriangles 'T1, ••• , 'Tn into overlapping triangles 'Tf, ... ,'T~ so that the area of their union, the Perron tree, is not more than 1. Those vertices formerly at the center of the square are now the tips of the branches of the Perron tree. Let an interval of length 1 be pivoted at the tip of 'Tf, rotated counterclockwise through the vertex angle of 'Tf, lifted and translated to the tip of 'T~, pivoted at the tip of 'T~, rotated through the vertex angle of 'T~, •••• The result is a (discontinuous) movement that keeps the interval always within the Perron tree when rotation takes place and effects a counterclockwise rotation of the interval through an angle of 90°. Repeated for T2 , • •• ,T4 the procedure effects a rotation of S through 360° within a figure of total area not exceeding four times the area of a Perron tree.

Section 2.2. Measure Theory

125

Figure 2.2.1.3. The building of a Perron tree.

As in Figure 2.2.1.4 the Pal join GLULMNUDL

permits the replacement of the discontinuous movement just described by a continuous movement. The triangles DEF and GHI are translates of typical adjacent subtriangles in the original square. Thus DF and GH are parallel. If f > 0 the point K is chosen so that

Then the interval S, say XY, lying on DE and with Y at Dis: i. pivoted at D, rotated until XY lies on DF (DF

c

DL)j

Chapter 2. Analysis

126

ii. iii. iv. v.

slid along D L until X is at L; pivoted at L, rotated until XY lies on LN; slid along GL until Y is at G; pivoted at G, and rotated until XY lies on G I. E

Figure 2.2.1.4. The Pal join. Thus S, lying in DE F U Pal join UGH I, is rotated through an angle equal to LEDI + LHGI, the sum ofthe angles oftwo adjacent subtriangles. Since the area of the Pal join is less than 8~ the area of the union of all Pal joins and of all Perron trees is not more than f

'2 + 4..\2 (Perron tree) . Hence if p > 1£6 it follows that carrying out this process in each of the large triangles T 1 , ••• ,T4 leads to a continuous 3600 rotation of the interval in a figure of area not more than f. Although the Pal joins permit continuous motions and do not significantly add to the area of the Perron trees, the diameter of the polygon produced via the Pal joins is significantly larger than the diameter of the original square T. The core of the Besicovitch solution is a systematic device for constructing a Perron tree of arbitrarily small area. The description that follows is drawn not from [Besl], in which the solution of the Kakeya problem first appeared, but from [Bes3], where the author's expository skill, accumulated over 35 years, is plainly evident. For p at least 3, lines parallel to and at heights !, 1, ... ,1 above the base of T1 are drawn. By recomposition-compressio::' {he decomposition of T1 into 2P - 2 subtriangles is successively reversed while compression is applied to yield versions of T1 that are similar but of heights ~. Furthermore, each recomposition-compression halves the number of subtriangles, cf. Figure 2.2.1.5, where the relation between the recomposedcompressed triangles and the decomposed original triangle T1 and its subtriangles Tb" ., Tn is shown. The purpose of recomposition-compression is

7"'"

Section 2.2. Measure Theory

127

simply to reduce to its most primitive form the operation of translating subtriangles for optimal overlap.

(

L__________________

I

/\

Tl

T2

TS

I T4

I

__~

21"<

Figure 2.2.1.5. Recomposition-compression (p

= 4,

n

= 24 - 2 = 4).

Then the procedure bisection-expansion illustrated in Figure 2.2.1.6 undoes recomposition-compression and leads to the creation of the Perron tree, a union of translates of the triangles ri. Notice that the area of the top end of each triangle ~k is always the area of ~b i.e., ~. In Figure 2.2.1.6 bisection-expansion is shown for the triangle rl of Figure 2.2.1.5. The "sapling" that "grows" into the Perron tree of Figure 2.2.1.3 is shown for the parameter values p = 4, n = 2P - 2 = 4.

Chapter 2. Analysis

128

triangle

bisection

bisection

expansion sapling

Figure 2.2.1.6. Bisection-expansion (p

expansion tree

= 4, n = 2P- 2 = 4). k+l k

k-l

L

M

L

T

P R

M

Figure 2.2.1.7. The basis of the area calculation. Because the quadrangle SQUV in Figure 2.2.1.7 is a parallelogram it follows that

Thus there obtains the equality

A2 (6.LQ R U 6.T S M)

= A2 (b:.LM N) + 2A2 (b:.NVU) .

Note also that in Figure 2.2.1.5 the area of the part (the "top end") of ~4 between levels 3 and 4 is the area of ~1. Hence in the bisection-expansion

Section 2.2. Measure Theory

129

illustrated above, the area of ~1 is increased by no more than 2A2 (~1). When bisection-expansion is applied p - 2 times to ~2 to produce the full Perron tree for the parameter values p and n = 2P- 2 , the area of that Perron tree does not exceed

Since Perron trees of arbitrarily small area can be constructed and since Pal joins of arbitrarily small area can be used, the Besicovitch solution is validated.

o Exercise 2.2.1.13. Show that when p = 3 the bisection-expansion procedure yields the optimal overlap of T~ and T2' i.e., the Perron tree when n = 2 has the minimal area achievable by overlapping translates of T1 and T2. In [Bes1] Besicovitch shows how his solution of the Kakeya problem yields as well the next result. THEOREM 2.2.1.6. THERE IS IN ]R2 A SET OF INTERVALS, ORIENTED IN ALL POSSIBLE DIRECTIONS, EACH OF LENGTH 1, AND THE UNION S OF THOSE INTERVALS IS A NULL SET: A2(S) = O. PROOF. When p ~ 3 the corresponding Perron tree is the union of translates TI of the constituent triangles Ti in T 1 • If Ti is to the right resp. left of the midpoint M of the base of T1 then Ti is translated to the left resp. right. Each line segment L from the vertex of T1 to a point X on the base of T1 is translated to a line segment within the Perron tree. Using p + 1 rather than p, Perronize the translate TI of Ti, •••• Each line segment L is translated to a sequence {Lp h::;p and the sequence {Xph::;p of base points converges to a point Xoo on the base ofT1 • Because the line segments Lp are translates of L the sequence {Lp h::;p converges to a line segment Loo. The set Sl of all such limiting line segments is a subset of each (mUltiply) Perronized figure and hence A2 (Sl) = O. On the other hand, because all Perronizations involve only translation, the set Sl consists of line segments oriented like the original line segments L of T1. If a similar process is applied to T2, T3, T4 the result is sets S2, S3, S4 and

is a set of line segments, each of length not less than 1 and A2 (S) = O. For each direction () there is in S a unique line segment oriented in the direction ().

o

Chapter 2. Analysis

130

If R is a Lebesgue measurable subset of JR2 and if every line in JR2 meets R in at most two points then, owing to Fubini's theorem, A2(R) = O. On the other hand, in Example 2.2.1.13 below there is defined a nonmeasurable set N meeting each line in at most two points. The idea is due to Sierpinski

[Si3]. Example 2.2.1.13. Let \11 be the first ordinal number corresponding to a well-ordered set of cardinality # (JR). Then

#[{o :

0

< \11}]

= #(JR).

Let 0 be the set of all open subsets of JR. Then, because JR2 is separable in its Euclidean topology, # (0) = # (JR) and so the set Fpos ~{

=

F: Fe JR, F

} =F, A2(F) >0

of closed subsets of positive measure in JR2 and S ~f {o : 0 < \11} are of the same cardinality, i.e., the sets of Fpos may be indexed by the elements in S: 0 +-+ Fa. In the set of all maps p of some initial segment

{o : 1:5 0 < (3 :5 \11 } ~f [1, (3) of S into the power set 21R of JR2 let P consist of those maps such that: i. p(o) E Fa;

ii. no three points in the range of p are collinear. Then P is nonempty, e.g., if (3 = 2 and p ( {I}) C Fl then pEP. The set 'R of ranges of maps in P may be partially ordered by inclusion. Zorn's lemma implies there is a maximal element R in 'R and for some initial segment [1, (3) and some q in P: q {[I, (3)} = R. If (3 < \11 then #(R) < # (JR) and there is a direction () different from that determined by every pair of points in R. Since F{3 E Fpos, Fubini's theorem implies that some line in the direction () meets F{3 in a set A of positive measure. Hence there is in A a point P{3 not collinear with any pair of points in R. Define q' according to: q

'(0) _ {q(O) P{3

if 0 < (3 if 0 = (3.

Then q' maps the initial segment [1, (3 + 1) into a set R' properly containing R, in contradiction of the maximality of R. Hence (3 = \11 and m. R meets every set Fa; iv. no three points in R are collinear. The set R C ~f JR2 \ R contains no set B of positive measure since such a set B must contain some Fa, hence must meet R. Fubini's theorem implies

Section 2.2. Measure Theory

131

that if R is measurable then A2 (R) = O. Hence if R is measurable so is R C and since R C contains no set of positive measure, A2 (R2) = 0, whence A2 (]R2) = 0, a contradiction. In other words, R is a nonmeasurable subset of]R2 and R meets every line in at most two points. Exercise 2.2.1.14. Let R be the set of Example 2.2.1.13. If x e ]R and the vertical line Vz through x meets R in one point (x, y) let f(x) be y. If Vz meets R in two points let f(x) be the larger of the corresponding ordinates. If Vz does not meet R let f(x) be 0, i.e., f(X)~f{max{y: (x,y)eR}

o

if{y:. (x,y)eR}~0 otherwise.

Let G be the graph of f. Show that at least one of G and R \ G is a nonmeasurable subset of ]R2. If G is measurable subset of]R2 let h be such that h(x) = {min{y : (x,y)eR}

o

if{y :.(x,y)eR}~0 otherwise.

Show that either the graph of f or the graph of h is a nonmeasurable subset of ]R2. Example 2.2.1.14. For n in Nand R a region in ]Rn the set R does not have content.

n Qn

Example 2.2.1.15. The region R bounded by the Jordan curve-image of Example 2.2.1.10. 117 does not have content since the measure of the boundary aR of R is positive. Example 2.2.1.16. For a positive, the compact set 8 ~f C~2 (cf. Example 2.2.1.9. 115) does not have content since A2 (a8) > O. If f is nonnegative and Riemann resp. Lebesgue integrable on [0,1) then def 8 = {(x, y) : 0 ~ y ~ f(x), x e [0, II} has (two-dimensional) content resp. is a Lebesgue measurable subset of]R2 and the two-dimensional content resp. two-dimensional Lebesgue measure of 8 is

11

f(x)dx.

By contrast there are the phenomena illustrated in Exercises 2.2.1.15, 2.2.1.16. Exercise 2.2.1.15. Let

XQn[O,I)

the and let 1/J be
+ 1.

Show

Chapter 2. Analysis

132

i. for x in JR, 4>(x) < t/J(x); ii. t/J - 4> is Riemann integrable on [0, 1] and the Riemann integral

10 is 1; iii. the set

1

(t/J(x) - 4>(x)) dx

S ~f { (x, y) : 4>(x) ~ y ~ t/J(x), x E [0,1] }

does not have two-dimensional content.

Exercise 2.2.1.16. Let E be a nonmeasurable subset of [0,1]. Show . def def that If 4> = XE and t/J = 4> + 1 then: z. for x in JR 4>(x) < t/J(x); zz. t/J - 4> is Lebesgue integrable on [0, 1] and the Lebesgue integral

10

1

(t/J(x) - 4>(x)) dx

is 1;

zzz. the set

S ~f { (x, y) : 4>(x) ~ y ~ t/J(x), x E [0, I]}

is not a Lebesgue measurable subset of JR 2 • 2.2.2. Measurable and nonmeasurable functions

Example 2.2.2.1. The Cantor function Co permits the definition of a continuous bijection \II : [0,1] 3 x 1-+ X + Co(x) E [0,2] (hence \11-1 is also a continuous bijection) that maps a Lebesgue measurable set of measure zero into a nonmeasurable set. Indeed, A[\II ([0, 1] \ Co)] = 1 whence A[\II (Co)] = 1 and so \II (Co) contains a nonmeasurable set E. On the other hand: A ~f \11- 1(E) C Co and so A is Lebesgue measurable; A(A) = 0; \II(A) (= E) is not measurable; since the continuous image of a Borel set is a Borel set it follows that A is a non-Borel subset of the Lebesgue measurable set Co of measure zero; v. in particular A is not an Fer; vi. there is no function f such that Discont(f) = A.

i. ii. iii. iv.

133

Section 2.2. Measure Theory

[Remark 2.2.2.1: Any two closed Cantor-like sets are homeomorphic (cf. LEMMA 2.2.1.1. 112). One may have measure zero and the other may have positive measure, cf. Example 2.2.1.3. 108·1 If I is a bounded measurable function and p is a polynomial then po I is measurable. The Stone- Weierstrafl theorem implies that if 9 is continuous (on a domain containing the range of f) then 9 0 I is measurable. Exercise 2.2.2.1. Let the notation be that used in Example 2.2.2.1. Show that although the characteristic function XA is measurable yet the composition XA 0 \11-1 is not measurable. A measurable function of a continuous function need not be measurable.

Exercise 2.2.2.2. Show that if I : IR 1-+ IR is monotone and 9 : IR 1-+ IR is measurable then both I and log are measurable. (Hence if h is a function of bounded variation then both hand hog are measurable.) The function XA resp. \11-1 of Exercise 2.2.2.1 is measurable resp. monotone but the composition XA 0 \11-1 is not measurable. A measurable function of a monotone function is not necessarily measurable. The following result is used often in a measure situation (X,S, 1'). THEOREM 2.2.2.1.

(EGOROFF) IF E E S, IF I'(E)

<

00,

AND IF

{fn}nEN IS A SEQUENCE OF MEASURABLE FUNCTIONS CONVERGING TO

I

A.E.

ON E THEN FOR EACH POSITIVE 6 THERE IS A SUBSEQUENCE

{fn.hEN AND A MEASURABLE SUBSET D OF E SUCH THAT I'(D)

In.

<6

~ ION E\D

[Halm, Roy, Rud]. If the sequence {fn}nEN in THEOREM 2.2.2.1 is replaced by an uncountable net {f~} ~EA the conclusion need not obtain.

Example 2.2.2.2. Let A be [0,1) ordered so that t1 t t2 iff t1 ~ t2' Then a net ZIt converges to ZI iff limt .....o ZIt = ZI. Let 8 be the nonmeasurable set in Example 2.2.1.1. 106. Let {rn}nEN be an enumeration of Q n [0,1) and define 8n to be (8 + rn)/Z. Then there is a positive 6 such that )..·(8)=6 (=>"·(8n ), nEN).

Chapter 2. Analysis

134

For t in I n ~ [2- n - 1 , 2- n ) let It be defined by the equation

It(x) Since [0,1)

= {I o

= UnEN8n,

(0,1)

if x E ~n and x otherwISe.

= 2n+lt -

1

= l:JnENJn , if t E (0,1) then

})={1o

#({X: h(X)=F O

if2n+l~_1E8n

otherwIse.

It follows that each It is a bounded measurable function different from zero for at most one x in [0,1) and that if x E (0,1) then limt-+o It (x) = O. In short, It ~. O. If A*(D) < 6 then for each n in N, 8 n \ D =F 0. Choose Xn in 8 n \ D. As t traverses I n , 2n+lt -1 traverses [0,1) and there is in I n a tn such that 2n+ltn - 1 = X n , whence It n (xn) = 1. As n -+ 00, tn -+ 0 and thus off D, It ;'0: Although limt-+o It (x) = 0 for each x in (0,1) there is in (0,1) no set D such that A*{D) < 6 and as t -+ 0, It{x) ~ 0 off D. If the hypothesis JJ(E) < 00 is dropped from Egoroff's THEOREM, again the conclusion fails to obtain.

Example 2.2.2.3. Consider the measure situation (N, 2N, JJ) in which JJ is counting measure. If for n in N, In is the characteristic function of the set {I, 2, ... , n} then on E ~f N, limn-+<Xl In is the constant 1. If 0 < f < 1 and JJ(D) < f then D = 0. However, {fn}nEN does not converge uniformly to 1 on N \ D (= N). Let (X,S,JJ) be a measure situation and let {fn}nEN be a sequence of measurable functions. There are defined several modes in which the sequence might converge to a function I. Convergence a.e.:

In ~. I

<:}

JJ [{ x

In(x)

+I{x) }l = o.

Convergence in measure:

In m~asl

<:}

{f > 0 => n-+<Xl lim JJ [{ x : I/n(x) - l(x)1 > ell = O}. Convergence in p-mean (when In, I are in V(X»:

Section 2.2. Measure Theory

135

Dominated convergence: (when p

In,

I

n E N and

are in V(X), 1

~

< 00):

f

n-In

doml

If p.(X)

<=>

~.

I

and there is in V(X) a 9 such that

< 00 then dom

{

dom

{

- =>

If p.(X)

I/nl ~ Igl.

= 00 then - =>

a.e. } -

IIJp

a.e. -

II-l'

meas

=> -

.

=> m~as

•

Exercise 2.2.2.3. Show that if In m~as I then there is a subsequence Ink ~. I· The Exercises that follow are designed to show that the implications above are the only valid ones relating the different modes.

{fnkhEN such that

Exercise 2.2.2.4. Show that if def

In(x)= then

fn

a.e. -

0b t U

f

{n0

II Ail O·

r

n

if 0 <

x< ~

ifxEIR\(O,~),nEN

a.e . ....,(.

,l.e., -

7'"

II 111 -

d a.e .....,(. an - 7'"

Exercise 2.2.2.5. Show that if n = 2k marching sequence {fn}nEN given by

+ m,

is such that In IIJ1 0 and In m~as 0 but for all II 111....,(. a.e. II IIp....,(. dom d meas....,(. a.e. 7'" - , 7'" - , an - 7'" - .

dom -.

<

0 ~ m

x in

[0,1],

In(x)

Exercise 2.2.2.6. Show that, in the notation above, if

then

fn

meas -

0b t U

f

n

II Ail O·

r

meas....,(.

,I.e., -

7'"

II 111 -

d meas....,(. an - 7'"

dom

2k then the

-.

-;40, i.e.,

Chapter 2. Analysis

136

Exercise 2.2.2.7. Show that if

In(x) d~f {I if n ~ : ~ n o otherwise then In ~. 0 but In

0: if I'(X)

m:ps

+ 1,

n E N

= 00 then ~.~

m~as.

Exercise 2.2.2.8. Assume a) Un}~=o C Ll([O, 1], "x), b) In ~ 0, n E m~as 10, and d)

N, c) In

11

In(x) dx

_11

lo(x) dx as n -

00.

Show that 10 E Ll([O, 1], -X) and that if E is a Lebesgue measurable subset of [0, 1] then lim f In (x) dx n_oolE

= lEf lo(x) dx.

[Hint: Exercise 2.2.2.3. 135 and THEOREM 2.2.2.1. 133 imply that if f > 0 then E contains a measurable subset EE such that "x (E \ EE) < f and

lim f In.(x)dx k-oolE,

=f

lE,

lo(x)dx.

Hence limsuPk IE In. (x) dx ~ IE lo(x) dx. Fatou's lemma implies

f

lE

lo(x)dx

~ liminf f k

If limn _

oo

lE

In.(x)dx

IE In(x) dx

~ limsup f k

k In. (x) dx ~ lEf lo(x)dx.

# IE 10 (x) dx

there is a subsequence

Un~ hEN and a positive 7] such that I IE In~ (x) dx- IE lo(x) dxl ~ 7], The preceding argument applied to Un' hEN yields a contra-

•

diction.]

Example 2.2.2.4. Show that if

go(x )

if x # 0 , = { o1x .Ifx=O

gn () X

= { nx1

ifO<x
I

, n E

N

n-

then for the "marching sequence" in Exercise 2.2.2.5. 135 and for properly chosen constants Cn , the sequence {gn + engnln d~f hn}nEN is such that a) {hn}nEN C Ll([O, 1], "x), b) h n ~ 0, n E N, c) h n m~ go, and d)

11

hn(x) dx -

00

=

11

go(x) dx

Section 2.2. Measure Theory

137

whereas if f E (0,1) then lim sup n--+oo

11

hn(x) dx

= 00 t=

11

go(x) dx.

E:

£

For contrast cf. Exercise 2.2.2.8. 136. The Radon-Nikodym theorem states that if X in (X, S, It) is u-finite and if v is a u-finite measure absolutely continuous with respect to " (v « It) then there is a nonnegative measurable function f such that for every E in S: v(E) = IE f(x) d".

Exercise 2.2.2.9. Show that if X ~f JR, S ~f 2R and v(E)

~f {~ if E is countable, finite, or empty

,,(E)

~f {~

otherwise if #(E) = n, n E N otherwise

(" is counting measure) then v «". Show there is no nonnegative measurable function f such that for every subset E of X v(E) =

Is

f(x) d".

Let (X, S, ,,) and (Y, 'R, v) be two measure situations and let H : X Y be a measurable map, i.e.,

1-+

Then on 'R there can be defined an image measure H" according to the formula:

Example 2.2.2.5. Assume S is the set C of Lebesgue measurable subsets of [0, 1], that 'R = {0, [0, and that H is the identity map. Then [.25, .75] ~ 'R, H- 1 ([.25, .75]) = [.25, .75] E S but

In,

H ([.25, .75]) = H [H- 1 ([.25, .75])] = [.25, .75] ~ 'R. In other words, H" is not definable for all images of sets in S. However, since H- 1 nO, 1] \ H([O, I])} = H- 1 (0) it follows that

H" nO, 1] \ H([O, I])}

= ,,(0) = 0,

Chapter 2. Analysis

138

Le., [0,1] \ H([O, 1]) is measurable and is a null set: The image measure HI' is concentrated on im(H). Associated with a measure situation (Z, T, p) are the outer measure p. and inner measure p•. Each is defined on 2z. In terms of the inner measure and outer measure it is possible to illustrate the image measure catastrophe in which the image measure HI' is not concentrated on im(H). Example 2.2.2.6. Let a subset X of [0, 1] be nonmeasurable and such that ,x.(X) = = 1 - ,x. (X). The inclusion map

°

H: X :3 x <-+ x E [0,1]

permits the definition in X of au-algebra S ~f H-l (X n C) and on S a measure 1': if A E S there is in C an 1 such that A = X n 1 and JL(A) ~f ,x (If BE C and A = XnB then (B\A)U(A \B) c [O,I]\X and so ,x(B) = ,x(A), Le., JL(A) is well-defined and the image measure HI' is definable.) However, although H-l ([O, 1] \ H(X)} is empty, [0,1] \ H(X) is not empty: H-l ([O, 1] \ H(X)} = 0 and [0,1] \ H(X) # 0. Furthermore, [0,1] \ H(X) is not measurable, and

(1).

,x* ([O, 1] \ H(X)} =

°

= 1 -,x* ([O, 1] \ H(X)} :

The image measure catastrophe has occurred: the image measure HI' is not concentrated on im( H). Exercise 2.2.2.10. Let {In} be the sequence of open intervals deleted from [0,1] in the construction of a Cantor set C of positive measure. For :$ a < b :$ 1 and n in N, let gn,J be any open interval J ~f (a, b), continuous, piecewise linear, and such that :

°

I

ifO<x
gn,J(x) = { 0 if a + b;.a :$

X

:$ b _

b;.a

1 ifb:$x:$1. For n in N let

In

be the product gn,h . gn,I2' ... . gn,In '

Show that: i. In! xc; ii. Xc is a bounded semicontinuous function that is not Riemann integrable; iii. there is no Riemann integrable function h such that h = Xc a.e.

Section 2.2. Measure Theory

139

[ Note 2.2.2.1: By means of bridging functions (cf. Exercise 2.1.2.6. 62) the functions gn,J. and hence also the functions In, may be replaced by infinitely differentiable functions while the conclusions above remain valid.] Exercise 2.2.2.11. Show that if C in [0,1] is a Cantor set of positive measure, (a, b) is the generic notation for an interval deleted from [0,1] in the construction of C, and

I(x) ~f {O ~f 0 ~ x < 1 11fx=1 def

g(x)

=

{I

1-

~(b -

a)

+ Ix - ~(a + b)1

if x E C if x E (a, b)

then I is Riemann integrable, 9 is continuous, and there is no Riemann integrable function h such that h = log a.e.

Exercise 2.2.2.12. Show that, in contrast to the results above, a continuous function of a Riemann integrable function is Riemann integrable: if I is continuous and 9 is Riemann integrable then log is Riemann integrable. Show also that if I is continuous and 9 is measurable then log is measurable (cf. Exercise 2.2.2.1. 133). [Hint: Assume first that I is a polynomial. Then use the Weierstrap approximation theorem.] Example 2.2.2.7. Assume

g(X)~f{x02Sin(~) ifx#:O

otherwise.

If c > 0 let Xc be sup {x : 0 to the rule

<x

~

c,g'(x) = O}. Then define gc according

x _ {g(X) if 0 ~ x ~ Xc gc( ) - g(x c) if Xc < x ~ c.

Let C in [0, 1] be a Cantor set of positive measure, let (a, b) denote the generic interval deleted from [0,1] in the construction of C, and let c be ~(b - a). For x in [a, b] define I according to the rule:

I(x)_{gc(x-a) ifa~x~~(a+b) - gc( -x + b) if + b) ~ x ~ b.

Ha

A direct calculation shows that I is differentiable on [0, 1] and that f' (x) = 0 on C. Since g'(O) = 0 and

g'(x) = 2xsin (~) - cos

U)

if x#: 0,

Chapter 2. Analysis

140

it follows that 1f'(x)1 ~ 3 on [0,1] and f' is discontinuous on C, whence is not Riemann integrable: The function f' is bounded, has a bounded primitive I and yet f' is not Riemann integrable nor is there a Riemann integrable function h such that h = f' a.e.

f'

Exercise 2.2.2.13. Show that if I(x) =

then

{1

8i x :

if x ¥ if x =

°°

I is continuous on JR, lim

[R I(x) dx

R-+ooJo

exists, but I is not Lebesgue integrable on [0,00). If, in (X, S, J.t), J.t(X) is not finite, the following pathology may arise. Exercise 2.2.2.14. Assume R,.. t 00. Find in JR sequences {an}nEN and {an} nEN of positive numbers such that an t 00, E:=l an < 00,

°

Note that In ! 0, In ~ 0, and II/nlll ! but the convergence to zero of II/nlll can be arbitrarily slow compared to the convergence of II In 11 00 , For the product measure situation (X x Y, S x T, J.t x /I) there are two important theorems, due to Fubini and Tonelli. THEOREM (FUBINI). IF (X, S, J.t) AND (Y, T, /I) ARE MEASURE SITUATIONS AND IF I: XxY f-+ JR IS MEASURABLE AND III IS INTEGRABLE WITH RESPECT TO J.t x /I THEN FOR ALMOST EVERY x RESP. Y THE RESTRICTED MAP

Y

f-+

JR

I(Y) : X

f-+

JR

(WHEN

x

IS HELD CONSTANT) I(x) :

(WHEN

Y

IS HELD CONSTANT)

RESP.

IS MEASURABLE AND

LxY

I(x, y) d(J.t x /I) =

L([

= [

l(x)(Y) d/l) dJ.t

(L I(Y)(x) dJ.t) d/l.

Section 2.2. Measure Theory

141

THEOREM (TONELLI). IF I IS NONNEGATIVE AND MEASURABLE, IF AND (Y,T,V) ARE CT-FINITE, AND IF EITHER OF THE ITERATED INTEGRALS

(X,S,J.L)

IS FINITE THEN I IS INTEGRABLE (WHENCE, BY FUBINI'S THEOREM, THE TWO ITERATED INTEGRALS ARE EQUAL AND THEIR COMMON VALUE IS

r

lxxy

I(x, y) d(J.L x v).)

The importance of the integrability of I in Fubini's theorem and the importance of the CT-finiteness assumption in Tonelli's theorem are revealed in the two parts of Example 2.2.2.8.

Example 2.2.2.8. i. For the measure situation ([0,1]2, £ x £, >. x >.) and for n in N, let h n be a continuous nonnegative function such that the support of h n is contained in In

~f (n~l' ~)

and f[o,l) hn(x) dx

= 1.

Then for each

(x, y) in [0,1]2, at most one term of the series 00

L (hn(x) - hn+l(x)) hn(y) ~f I(x, y)

n=l

is not zero, I/(x, y)1 = E:=llhn(x) - hn+1 (x)llhn(y)l, and I is continuous except at (0,0). Hence I is measurable. Furthermore,

r

I/(x, y)1 d (>. x >.)

lIn xIn

r (r

l~q A~q

= 1,

r

1[0,1) x [0,1)

I(X,Y)dY) dx = 1 ¥ 0=

I/(x, y)1 d (>. x >.)

r (r

A~q A~q

= 00

I(X,Y)dx) dy.

Thus, absent the integrability of III, the conclusion of Fubini's theorem cannot be drawn. ii. For a measure situation (X, 2x , J.L), J.L is counting measure iff whenever 8 c X then J.L(8) = {#(8) if 8 is ~nite 00 otherw1se. Assume that in the measure situation ([0, 1],2[0,1), J.L) J.L is counting measure (whence [0,1] is not CT-finite) and consider the measure situation ([0,1] x [0,1],2[0,1) x (£ n [0,1]), J.L x >.). Assume

B ~f {(a, a) : a E [0, I]}.

142

Chapter 2. Analysis Then, B(x) resp. B(Y) denoting the set of y resp. x such that (x, y) E B,

In other words,

r [r 1

10

1[0,1)

XB(X, y) dJLl dA

=0

and

r [r XB(X, y) dA] dJL = 1, 1

1[0,1) 10

i.e., both iterated integrals exist but are unequal even though XB is a bounded nonnegative 2[0,1) x (.c n [0, l])-measurable function. Thus, absent the O'-finiteness condition, the conclusion of Tonelli's theorem cannot be drawn.

Exercise 2.2.2.15. In the context of Example 2.2.2.8 above, B is the graph of the measurable function f : [0,1] 3 x 1-+ x whence B is JL x A-measurable. Show JL x A(B) =

r

XB(X, y) d(JL x A) =

00.

1[0,1) x [0,1)

Example 2.2.2.9. Let R in IR2 be a nonmeasurable subset that meets every line in at most two points (cf. Example 2.2.1.13. 130). Then XR is nonnegative and not measurable whence

does not exist but

l (l

XR(X,Y)dX) dy

=

l (l

XR(X,Y)dY) dx

= O.

[Remark 2.2.2.2: In Example 3.1.2.5. 193 there is described a set r that is dense in IR2 and meets every horizontal resp. vertical

Section 2.2. Measure Theory

143

line in exactly one point. Let r 1 be r n [0, 1]2. Then the Riemann double integral IrO.l]2 Xr 1 (x, y) dA does not exist although

[

llo.l]

([

llo.l]

Xr 1 (x, y) dX) dy

Exercise 2.2.2.16. For

I

=[

llo.l]

([

llo.l]

Xrl (x, y) dY) dx

= 0.]

in lRlR and a in lR define the sets

8<0 ~f 1-1 ((-oo,aj) 8<0!. ~f

1-1 ((-oo,a)) 8>0!. ~ 1-1 ([a, 00)) 8>0!. ~f

r

1

((a, 00)).

Show that I is (Lebesgue or Borel) measurable iff for all a, 8~0!. is measurable, iff for all a, 8<0!. is measurable, iff for all a, 8?0!. is measurable, and iff for all a, 8>0!. is measurable.

Exercise 2.2.2.17. Let E be a nonmeasurable subset oflR. Show that if id denotes the map lR 3 x 1-+ X E lR and

I dd·d = XE . 1 then

I

XIR\E . 1·d

is nonmeasurable although for every a in lR

consists of at most two points and hence is measurable. 2.2.3. Group-invariant measures Let 8 be a set, let 80 be a fixed nonempty subset of 8, and let G be a group of autojections, i.e., bijections of 8 onto itself. The problem to be considered is that of determining whether, on the power set S ~f 28 , there exists a finitely additive measure It such that: i. 1t(80 ) = 1 (It is normalized); ii. if 9 E G and A E S then It (g(A)) = It(A). Such a It is called an [8,80 , G]-measure and is an instance of a groupinvariant measure.

Chapter 2. Analysis

144

Example 2.2.3.1. For 8 an arbitrary set, 80 a finite subset of 8, and G the set of all bijections of 8 onto itself, assume # (80 ) = n (E N). Define I" as follows: I"(A) ~f {~#(A) if #(A~ E N U {OJ 00 otherwIse. Then I" is an [8,80 , G)-measure. In other words, if 8 0 is finite and I" is an 8 0 -normalized counting measure then, for any group G of autojections of 8, i-ii are satisfied. Let a group G be called measurable if there is a [G, G, G)-measure. In [Nl] von Neumann showed that:

i. every abelian group G is measurable; ii. if H is a normal subgroup of G and if both Hand G / H are measurable then G is measurable. Thus "measurability" is, in the current context, a QL property. In particular, for n in N, Qn, R. n, and Tn are measurable groups. [ Note 2.2.3.1: It must be noted that the measures with respect to which abelian groups are measurable are not necessarily count ably additive. On the one hand, counting measure, which is a count ably additive measure, is, for any countable group, abelian or not, automatically a measure with respect to which the group is a measurable group. However, if G = T then the group-invariant measure, say 1", derivable from von Neumann's result cannot be count ably additive. Indeed, if I" is count ably additive it is, in particular, a nontrivial translation-invariant count ably additive measure on the O'-ring S(K) generated by the compact subsets of T. Thus I" is Haar measure and, according to the results in Subsections 1.1.4 and 2.2.1, there is in T a set 8 such that 1"(8) = 0 = 00, a contradiction. Similar observations apply to R.n.) On the other hand, Hausdorff [Hau] showed that if Grigid is the group of rigid motions of 1R? and (x,y,z )

x 2 + y2 + z2 :5 1 }

8 1 ~f {(x,y,z)

X 2 +y2+Z2=1}

Bl

=

def {

(=8Bd

are the unit ball and the surface of the unit ball of R.3 then there is no [R.3 , 81. Grigid] -measure. Consideration of unions of spherical shells reveals that there is no [R.3 , B1. Grigid] -measure. Hausdorff's result is consonant with von Neumann's because the group Grigid of rigid motions of R.3 contains the subgroup 80(3) of all rotations about axes through the origin 0 of R. 3 , and as the next lines show among other things, 80(3) is not abelian. The group 80(3) is isomorphic, according to the maps described next, to the multiplicative group HI of quaternions of norm 1.

Section 2.2. Measure Theory

145

Example 2.2.3.2. The correspondence def 1 + b'l+CJ+ • dk q=a

(~ ~) + b (~ ~) + c (~ ~1) + d (~ ~i)

-

a

=

(a.++ c

di

b~

bi -

a-

~) ~f

d~

(af3 -!) ~f a

A . q

is an isomorphism between lHl and a subalgebra of the algebra Mat22 of 2 x 2 matrices over C. Furthermore, if q E lHll then Aq is a unitary matrix. Let Coo denote the extended complex plane with the "point at infinity" 00 adjoined. The map

TA

: q

Coo 3 z 1-+ {

~:~~ if f3z + 0 # 0, z # 00 if f3z + 0 = 0, Z # 00 ~ if Z = 00, f3 # 0 00 if Z = 00, f3 = 0 00

is an auteomorphism of the extended plane Coo. The association TAq - Aq is a group isomorphism. The standard stereographic projection of Coo o~ the Riemann sphere S t converts the map T Aq into an auteomorphism T Aq of Sl. Every auteomorphism T of SR has a fixed point. [PROOF: Ifx E SR 2 the sequence {Tnx}nElII has a limit point y and Ty = y.] The corresponding fixed point of T Aq is a solution , of the equation

,-1

f3z2 + (0 - a)z + 73 = o.

Furthermore, is a second solution corresponding to a fixed point. The stereographic images of these fixed points are diametrically opposite points of Stand T Aq is a rotation about the axis through them. In this way lHll is isomorphic to the set ofrotations of St, i.e., to the set ofrotations of SI or of 1R3 . Since lHll is not abelian neither is SO(3) nor Grigid abelian. An important consequence of Remark 1.1.5.1. 17 is that lHlI. i.e., SO(3), contains a free set of cardinality # (1R). This fact is basic to the derivation of the Banach-Tarski "paradox" to which the remainder of this Section is devoted. Call two subsets A and B of 1R3 congruent if there is a rigid motion U such that U(A) = B. Hausdorff's idea was exploited by Banach and Tarski to show that, ~ denoting "congruent," the ball Bl in 1R3 can be decomposed into m pieces CI..'" Cm such that

into n pieces Cm+I."" Cm + n such that

Chapter 2. Analysis

146

into m

+n

pieces At, . .. ,Am+n such that

and such that

[BanT]. Thus B1 can be decomposed and the pieces can then be reassembled via rigid motions to form two balls, each congruent to B 1 • This theorem was polished and refined by Sierpinski, von Neumann, and finally by Robinson to yield the following result. THEOREM 2.2.3.1. IN THE UNIT BALL B1 OF a3 THERE ARE FIVE PAIRWISE DISJOINT SETS, A 1, ... , As, THE LAST A SINGLE POINT, AND B1 = A 1l:J···l:JAs B1 ~ A 1l:JA 3 ~ A2l:JA4l:JAs·

The ingredients of the proof of THEOREM 2.2.3.1 are straightforward and are assembled below in a pattern based on Robinson's development [Robi]. Except at the very end, where a single translation is invoked, only rotations of a 3 are used for the rigid motions that establish the relevant congruences. Reflections are not used. At first the focus is on the decomposition of the surface Sl ~ 8B1 of the unit ball B1 and in that discussion the only rigid motions used are rotations. The goal is to show that there are two different decompositions of Sl: Sl = A 1l:JA 2 l:JA 3l:JA 4 Sl

= C1l:JC2l:JC3l:JC4l:JCS

(2.2.3.1)

such that A1

~

A2

~

A 1l:JA 2

A3 ~ A4 ~ A3l:JA4

(2.2.3.2)

C 1 ~ C 2 ~ C1l:JC2 l:JCS C3 ~ C4 ~ C3l:JC4.

(2.2.3.3)

In the second decomposition, Cs is a single point P. Furthermore, there are for SR, 0 < R < 1, decompositions analogous to that in (2.2.3.1) and with properties analogous to those in (2.2.3.2)-(2.2.3.3).

Section 2.2. Measure Theory

147

Associated with a finite decomposition {At. ... , An} of 8 1 and a congruence

is a canonical relation R having domain and range N ~f {l, ... , n} and such that iRj iff i E K ~f {kt. ... , kr } and j E L ~f {It, ... , Is}. A rotation U is compatible with the congruence if no point of U (Ai) lies in Aj unless i E K and j E L, i.e., U is compatible with the corresponding relation R if no point of U (Ai) lies in Aj unless iRj: U (Ai) n Aj :F 0 ~ iRj. Any relation in N is defined by a subset 'R, of N x N: iRj ¢} (i, j) E 'R,. Hence without regard to congruence, one may speak of a relation Rand its corresponding subset 'R, of N x N: R '" 'R,. The discussion below is confined to those relations R having domain and range N. In other words each image of the two projections of 'R, onto the factors of N x N is N. If 8 1 = l:Jf=1 Ai then for any relation R and any rotation U the notion of their compatibility remains unchanged. If Rl and R2 are relations their product RIR2 is the relation R3 such that iR3k iff there is a j such that iRd and jR2k. The inverse R-l of a relation R is characterized by the statement: iR- 1 j iff jRi. If iRi then i is a fixed point for R. The identity relation ~d corresponds to the "diagonal" A U

def { ( t,) . . ) : ,. = )' } ' D.' • • = : '-'Lid) ¢} Z =).

If R '" N x N then R = R- 1 and so RR- 1 = R :F ~d, i.e., the product of a relation and its inverse need not be the identity relation.

Exercise 2.2.3.1. Show that if U and R are compatible then (since U has a fixed point) R has a fixed point. Exercise 2.2.3.2. Show that if Ui and ~ are compatible, 1 ~ i then U1 ••••• Urn is compatible with Rrn ..... Rl'

~

m,

[Hint: Note the reversal of order in the product of relations. Use induction.]

For a free set {Ut. ... , Urn} of rotations of 8 1 let G be the group generated by them. Then each element of G is uniquely representable as a reduced word UiE11 ••• Ui; , i.e., a word that does not simplify (cf. Exercise 1.1.5.1. 9). If x E 8 1 then Gx ~f {U(x) : U E G} is the orbit or trajectory of x. A point x in 8 1 is a fixed point if, for some U in G and not the identity id of G, U(x) = x. As an auteomorphism of 8 1 each U in G has a fixed point z. Since U E [1R3 ] it follows that the antipodal point -z is also a fixed point for U, just as physical intuition suggests. Since det(U) = 1 and all the eigenvalues of U are in T, it follows that if U :F id then the eigenvalues of U are, for some ( in T, {l, (, ().

Chapter 2. Analysis

148

Exercise 2.2.3.3. Show: a) that a trajectory consists entirely of fixed points or contains no fixed points; b) two trajectories are either disjoint or coincide; c) Sl is the (disjoint) union of the trajectories. [Hint: If x is fixed for U then V(x) is fixed for VUV- 1.] Exercise 2.2.3.4. Show that if T is a trajectory without fixed points and x E T then for each y in T there is in G a U such that y = U(x). A trajectory consisting of fixed points may be described in a manner similar to that in Exercise 2.2.3.4 although the details of the description, given next, are more complex. Let T consist entirely of fixed points. Among all rotations having fixed points in T there is at least one, say W, for which the corresponding reduced word is shortest. Assume W(x) = x. Exercise 2.2.3.5. Show that the first and last factors of W are not inverses of each other. Thus Wand W-1 do not begin with the same factor nor end with the same factor. [Hint: Otherwise, for some rotation V, V- 1WV has a fixed point in T and V- 1WV, reduced, is shorter than W.] LEMMA 2.2.3.1. IF V(x)

=x

THEN FOR SOME n IN Z, V

= Wn .

PROOF. Since Wand V have the same fixed point, they are rotations around the axis through x and hence they commute: WV = VW. Hence V = WVW-1. If WV does not simplify, then the unique representation of V begins with the block W. Hence for some n in N, V = wn Z and Z does not begin with W. However, V = wnvw- n = w2n zw- n whence wnzw-n = Z, and so V = wn Z = zwn, which does not begin with W. If zwn simplifies then, since V begins with W, V = W n - k , k > 0, a contradiction. Hence zwn does not simplify and so Z = id and V = W n. If WV does simplify then, owing to Exercise 2.2.3.5, W- 1V does not simplify and the previous argument shows that for some n in N, V = W- n .

o Exercise 2.2.3.6. Show that if yET then for some X that does not end with W nor with the inverse of the first factor of W, y = X(x). Show also that such an X is unique. [Hint: For some Z, y = Z(x) and if Z ends with W, then y = Z(x) = YW(x) = Y(x). After finitely many steps, y = X(x) and X does not end with W. If X ends with the inverse of the first for large enough n. factor of W, consider

xwn

Section 2.2. Measure Theory

149

If y = X(x) = X'(x) while X and X' are as described, then X-IX' fixes x and so X-IX' = wn, n E Z. If n > 0 then X' ends with W. If n < 0 then reverse the roles of X and X'. Hence n = 0.]

The next step in the argument is the derivation of the connection between a set of relations and the possibility of decomposing 8 1 in a manner associated to the relations. For this purpose the algebra described above for relations is quite useful. THEOREM 2.2.3.2. LET Rl"'" Rm BE RELATIONS FOR WHICH N IS BOTH DOMAIN AND RANGE. THEN 8 1 CAN BE DECOMPOSED INTO n PIECES Ab"" An AND FOR THIS DECOMPOSITION THERE ARE ROTATIONS Ub" ., Urn COMPATIBLE RESPECTIVELY WITH R l , • •. , Rm IFF EACH PRODUCT OF FACTORS OF THE FORM R:, f = ±l, HAS A FIXED POINT. FURTHERMORE, IF SUCH ROTATIONS EXIST THEY MAY BE CHOSEN TO BE A FREE SET IN 80(3). PROOF. If 8 1 = U~=IAi and if rotations Ui as described exist and R ~f R:: ... R:; is given then U ~f U iE: ••• UiE1l is compatible with Rand since U has a fixed point so does R, cf. Exercise 2.2.3.3. 148. Conversely, assume every R as described has a fixed point. Choose m free rotations, Ub" ., Urn. The next argument uses the results in Exercises 2.2.3.4. 148 and 2.2.3.6. 148. The task is to define a decomposition {Ab .. . , An} of 8 1 so that, for the free set U ~f {Ub ... , Urn} of rotations, each Ui is compatible with the corresponding R;. Since the group G generated by U is countable G may be enumerated systematically so that first only rotations (reduced words) that have exactly one factor are listed, then those having only two factors, . .. . Let Vo be id and let the enumeration of G \ {id} be Vn , n EN. Throughout what follows the fundamental assumption that the domain and range of each R; is N proves essential. Case 1. Assume the trajectory T has no fixed points. Let x be a point in T. Start the construction of Al by the declaration: x E AI' If VI = Uti, then since there is an I such that lR:il, start the construction of A, by the declaration: Vl(X) E A,. Note that Al n A, = 0. Having constructed or made assignments to pairwise disjoint sets already constructed for all reduced words having at most n factors, assume VM+l ~f U?VM is the first word having n + 1 factors. If VM(X) E Ak, there is a p such that kRjj p. If Ap has been constructed, assign VM+l(X) to Ap. Otherwise construct Ap by the declaration: VM+l(X) E Ap. By definition, Ap is disjoint from all sets Ai already in existence. The inductive procedure described above defines pairwise disjoint sets AI, . .. for a given trajectory without fixed points. The procedure is in-

Chapter 2. Analysis

150

dependent of the trajectory and thus the sets A b ... are defined for all trajectories having no fixed points. Case 2. Assume the trajectory T consists of fixed points. According to the earlier discussion, for a x in T, there is a rotation X such that every y in T is uniquely of the form X(x), and the rotation X ends neither with W nor with the inverse of the first factor of W. Let the reduced form of W be n:=1 Thus the points

uZ:

j

•

8

x,

U::· (x), ... , II U~~j (x) =

X

i=1

form a closed cycle. Once the points of the cycle have led to constructions or assignments to sets Aq the other points of T lead to constructions or assignments following the procedure in Case 1. Note that the hypothesis concerning the existence of a fixed point for every product of factors has not yet been invoked. Now the hypothesis is used to conclude that n:=1 R~: ~f R has a fixed point. Thus there are

R:

integers k o, ... , k8 such that kr-1Rtkr, 1 ~ r ~ s, and ko = k8 ~f k. If Ak exists, assign x to Ak. Otherwise declare Ak to consist of x. Similarly, for the other points of the cycle assign them to, or construct by declaration for them, sets Akr • Since the sets AI. . .. are pairwise disjoint and since every point of 8 1 is on some trajectory, it follows that 8 1 = l:Jl=IAi. Since the domain and range of each relation is N it follows that ? = n.

o LEMMA 2.2.3.2. U(Ad i' A 2 •

IF 8 1

=

A 1 l:JA 2 AND U IS A ROTATION THEN

PROOF. As a rotation, U is an auteomorphism of 8 1 and has a fixed point v. Assume v E AI. Then U (v) E Al \ A 2 •

o Let A b ... , An be pairwise disjoint subsets of 8 1 and assume 8 1 = l:Jf=IAi. Then {A b ... , An} is a finite decomposition of 8 1. The set of all congruences, of which a typical one is (2.2.3.4) is decomposable with respect to an equivalence relation == defined as follows. Let K be {kb ... ,kr }, L be {h, ... ,18}' and denote a congruence such as Then: (2.2.3.4) by

ct.

. CK -CN\K L = N\L; .. CK - CK\L. u. L = L\K'

z.

Section 2.2. Measure Theory

iii. MeN

151

=> cf == cf~t!.

Furthermore cf == Ct.' if there is a finite chain of congruences linked by == and of which cf is the first and cf,' is the last. An equivalence such as i is an equivalence by complementation and an equivalence such as ii or iii is an equivalence by transitivity. For the most part, the argument below is concerned with canonical relations tied to congruences, but the intermediate results are more easily described with respect to relations that are not necessarily canonical. If R is a relation on N, if kEN, and if 'R::J {(I, k), ... , (n, kn

then, by abuse of language, R is said to contain a constant (the constant relation Rk, by further abuse of language, the constant k).

Exercise 2.2.3.7. Show that: i. if R contains the constant k and if {(k, In C 8 then R8 contains the constant 1; ii. if R contains a constant then R has a fixed point; iii. if n ~ 3 there are two canonical relations Rand 8 such that R8 is not canonical (hence there are noncanonical relations); iv. if Rand 8 are canonical relations then R8 contains a constant or R8 is itself canonical; v. if R '" (K, L) then R has a fixed point iff [K

n L] U [(N \

K)

n (N \

L)]

i' 0.

[Hint: Ad iv: It suffices to consider the product of two canonical relations and then to proceed by induction. Assume kR1S {:} (k E K1 {:} s E Ld 8R2l {:} (8 E K2 {:} 1 E L 2). Show that if K1 = L1 or K1 = L2 then R1R2 is canonical. Show that if K2 n L1 i' 0 and K2 n (N \ L 2) i' 0 then R1R2 contains a constant. Argue similarly if K2 n L1 i' 0 or K2 n (N \ L 1) i' 0.] The contents of THEOREM 2.2.3.2. 149 can be translated into a statement about congruences, complementary congruences, and congruences arising from transitivity. THEOREM 2.2.3.3. THE SURFACE 8 1 MAY BE DECOMPOSED INTO n PIECES SATISFYING A GIVEN SYSTEM C OF CONGRUENCES IFF: i. NONE OF THE CONGRUENCES IN C IS A CONGRUENCE OF TWO COMPLEMENTARY SUBSETS OF 8 1 ; ii. NONE OF THE CONGRUENCES IN C IS EQUIVALENT (==) TO A CONGRUENCE OF TWO COMPLEMENTARY SUBSETS OF 8 1 .

Chapter 2. Analysis

152

PROOF. Since complementary subsets of 8 1 cannot be congruent (cf. LEMMA 2.2.3.2. 150) the necessity of Hi follows. The proof of sufficiency of the conditions rests on the conclusion of Exercise 2.2.3.7iv. 151: EITHER the product R of two canonical relations contains a constant, whence R has a fixed point, OR

R is itself canonical. When a product of two canonical relations is itself canonical, say Rl '" (Kb L 1), R2 '" (K2' L 2), R1R2

~f R '" (K, L).

let the superscript * on a subset A of N denote either A itself or N \ A. Then kRI means there is in N an s such that

Lr

(k,s) E K; x (s,l) E K; xL; (k,l) E K* x L*

Lr n K; ¥- 0.

(2.2.3.5)

(Note that there are sixteen such sets of conditions.) Each corresponds to the equivalence (==) of the congruence corresponding to R and the congruence corresponding to Rl or to R 2. One of the conditions (2.2.3.5) serves as the transitivity or complementation from which the cited equivalence can be inferred. When the product of canonical relations is itself canonical its associated congruence is equivalent (==) to the congruence associated to one of the factors in the product. If the product R contains no fixed point then R does not contain a constant and hence R is canonical. Thus, in the notation used above, [K

n L] U [(N \ K) n (N \ L)] = 0

whence K = N \ L and so R corresponds to a congruence of complementary subsets of 8 1 , i.e., R corresponds to a congruence equivalent to one of the congruences in the original system, contrary to the hypothesis of THEOREM 2.2.3.3. 151.

o Example 2.2.3.3. Let n be 4 and let C be the system

Ai A3

~

~

A2 A4

~

~

A 1 l:JA 2 A 3l:JA 4.

Section 2.2. Measure Theory

153

Then the only congruences equivalent via complementation and/or transitivity are the following: Al ~

A3

~

A2 A4

~ All:JA2 ~ A l l:JA 2 l:JA 3 ~ All:JA2l:JA4 ~

A3l:JA4

Hence there exist rotations Ui , I

~ All:JA3l:JA4 ~ ~

i

~

A 2l:JA 3l:JA 4.

4, such that

Ul(A l )l:JU3(A 3) ~ All:J·· ·l:JA4 = 8 1 U2(A 2)l:JU4(A4) ~ All:J·· ·l:JA4 = 81, i.e., two copies of 8 1 can be made from 8 1 itself. Example 2.2.3.4. In Example 2.2.3.3 choose a trajectory T consisting of nonfixed points and choose a point P in T. Define a new decomposition of 8 1 by assigning P to Cs ~ {P} and assigning Ui(P), I ~ i ~ 4, according to the following pattern:

Ul(P) ....... C3 or C4, U1l(P) ....... C l U2(P) ....... C3 or C4, U;l(p) ....... C2 U3(P) ....... C l or C 2, U;l(p) ....... C l or C2 or C 4 U4(P) ....... C l or C 2, Uil(P) ....... C l or C2 or C3 (81 = C l l:JC2l:JC3l:JC4l:JCS ). (Notice the considerable flexibility in the assignments above.) For any other point Q in T make assignments according to the algorithm in Case 1 of the proof of THEOREM 2.2.3.2. 149. The (canonical) relations to be observed are precisely those listed next:

Rl '" 'R 1 ~f {I} x {1,2} R2 '" 'R 2 ~f {2} x {I, 2} R3 '" 'R3 ~f {3} x {3,4} R4 '" 'R4 ~f {4} x {3,4}. The corresponding congruences are

C l ~ C2 ~ C l l:JC2l:JCS C3 ~ C4 ~ C3l:JC4 and then

Ul (Cdl:JU3(C3) ~ 8 1 U2(C2)l:JU4(C4) ~ 8 1 •

Chapter 2. Analysis

154

IfO < r < 1 let Sl(r) be { (x, y, z) : x 2 + y2 + Z2 = r2 } and, following the patterns in Examples 2.2.3.3 and 2.2.3.4, decompose them as follows: Sl(r) Sl

= A1(r)l:JA 2(r)l:JA3(r)l:JA4(r), = C1l:JC2l:JC3l:JC4l:JCS

0< r < 1

(= C 1l:JC2l:JC3l:JC4l:J{P}). Let Ak be Ckl:JUO
U1 (A~) = U2 (A~) = A~ l:JA~l:JA~ U3 (A~) = U4 (A~) = A~l:JA~ U3 (A~l:J{O}) = U4 (A~l:J{O}) = A~l:J{O}l:JA~. It follows that B1 is the union of five disjoint subsets:

(2.2.3.6) and if Ai ~f A~, i

= 1,2,4,5,

A3 ~f A~l:J{O}, then

U1(Ad = U2 (A 2 ) = A1l:JA2l:JAs U3(A 3) = U4(A4l:J{O}) = A3l:JA4.

(2.2.3.7) (2.2.3.8)

From (2.2.3.7) and (2.2.3.8) it follows that

A1 A3

9!! 9!!

(2.2.3.9) (2.2.3.10)

A1l:JA2l:JAs A3l:JA4

whence

From (2.2.3.7), (2.2.3.8), and the fact that As

9!!

{O} it follows as well

that

D [ Note 2.2.3.2: One of the pieces of B1 is As, i.e., the set {P} consisting of the single point P. Since the congruence {O} 9!! As cannot be achieved by a rotation around 0 but can be achieved by a translation, one of the rigid motions employed is a translation, not a rotation.]

Section 2.2. Measure Theory

155

THEOREM 2.2.3.4. IT IS IMPOSSIBLE TO FIND FOUR PAIRWISE DISJOINT SETS AND FOUR isometric MAPS

SUCH THAT

B1 = K 1l:J···l:JK4 B1 = T1 (Kd U T2 (K2) B1 = T3 (K3) U T4 (K4) .

(2.2.3.11) (2.2.3.12)

PROOF. (Note that the sets Ti (Ki) are not assumed to be pairwise disjoint and that the Ti are not assumed to be sense preserving.) If each Ti does not move 0, assume 0 E K 1. Then T1 (0) = 0 and, since each Ti , as an isometry, is perforce one-one, 0 ¢ T3 (K3) U T4 (K4) = B 1. In short, if no Ti moves 0 then B1 in (2.2.3.11) or in (2.2.3.12) lacks a center, a contradiction. Thus it may be assumed that T4 (O) =I O. Since B1 \ T4 (B 1) covers more than a hemisphere of 8 1 (cf. Exercise 2.2.3.8 below) and since T3 (K3) covers B1 \ T4 (B 1) it follows that T3 (K3) covers more than a hemisphere of 8 1 . Furthermore, T3(O) = 0 since otherwise, each of T3 (Bd and T4 (B 1) failing to cover a hemisphere of 8 1 (cf. Exercise 2.2.3.8 below), their union cannot cover 8 1 ; a fortiori, T3 (K3) U T4 (K4) ¥B1. a contradiction. Since T3- 1 is an isometry and T3- 1(O) = 0 it follows that Til (81 ) = 8 1 . If H is a hemisphere of 81. let P be its center, e.g., the "North Pole." Since Til is an isometry it carries every spherical triangle having a vertex at P and contained in H into a congruent spherical triangle. Hence K3 (= T3- 1 (T3 (K3))) itself covers more than a hemisphere of 8 1 • Hence K 1l:JK2l:JK4 and, a fortiori, K 1l:JK2 covers less than a hemisphere of 8 1 . It follows that T1 (Kd U T2 (K2) cannot cover 8 1 , much less B 1.

o

Exercise 2.2.3.8. Show that B1 \ T4 (B 1) covers more than a hemisphere of 8 1 .

[Hint: It may be assumed that for some positive a, T4 (O) (a, 0, 0). Hence if (x, y, z) E T4 (81 ) n 8 1 then x = ~.j

=

For an extensive treatment of measure-theoretic paradoxes see [Wag] where there is a treatment of the next generalization, proved also in [BanT, Strj, of THEOREM 2.2.3.1. 146.

Chapter 2. Analysis

156

Let A 9!' B mean that A and B are equivalent under the group of rigid motions and reflections. If X and Yare two bounded subsets of JR3 and if both their interiors are nonempty: X O , yo =f 0, then they are equivalent by finite decomposition, Le., X = X1l:J···l:JXm Y = Y1l:J·· ·l:JYm Xi 9!' Yi, 1:5 i :5 m.

Thus a pea can be decomposed into finitely many pieces that can be reassembled to form a body as large as the Sun. 2.3. Topological Vector Spaces

2.3.1. Bases Zorn's lemma implies that for every Banach space, indeed for every vector space, there is a Hamel basis. On the other hand, a complete orthonormal set (CON) in a separable Hilbert space is a Schauder basis or simply a basis in the sense given next. A subset 8 ~f {b~hEA of a Banach space V (endowed with a norm II II) is a (Schauder) basis (for V) iff for every x in V there is in CA a unique set {X~hEA such that x = E~EA x~b~, i.e., there is a countable or finite subset S of A and if f > 0 there is in S a finite subset S( f) such that IIx - E~ES(E) x~b~1I < f. Note that 8 need not be countable. For example, if

v~f{{at}

: atEc,L:latI2
Le., if V is a nonseparable Hilbert space, and if, for t in JR, def {

bt =

Db

}

sEIR

then the set of all b t is a complete orthonormal set for V and is a (Schauder) basis for V. If V is finite-dimensional the notions of Hamel basis and Schauder basis for V coincide. Exercise 2.3.1.1. Show that if V is infinite-dimensional no Hamel basis is a Schauder basis. [Hint: It may be assumed that each element hI' of a Hamel basis H (in a Banach space) is of norm one. Consider <Xl

L: 2-nhl'n n=l

~f x

Section 2.3. Topological Vector Spacp.s

157

for which there must be a finite representation E" x"hw] From this point on, basis means Schauder basis. In every infinitedimensional Banach space V there is an infinite-dimensional closed subspace U for which there is a basis [Ban, Day, Ge3]. If a Banach space V has a countable basis then V is separable. Only after many years of study by many mathematicians did Davie and Enfto [Dav, En£) show that the converse statement is false. They constructed in C ([0, 1],R.), the set of R.-valued functions continuous on [0,1] and topologized by the norm,

II

1100 : C ([0,1], R.) 3 f

t-+

sup If(x)l, zE[O,l]

a closed subspace for which there is no basis. A basis S ~f {Sn}nEN for C ([0,1], R.) was constructed by Schauder. The functi{\ns in S are piecewise linear and var(sn), the variation of each Sn, is finite. Indeed, var(sn) :5 2, n E N, i.e., the Schauder system is of uniformly bounded variation. Subsequently Franklin designed a sequence :F ~f {Fn}nEN of functions constituting a basis for C ([0,1], R.) [KacSt]. With respect to the standard inner product, (/,g) ~f 1 f(x)g(x)dx, the functions in :F are orthonormal. Since there is a basis for C ([0, 1], R.), the DavieJEnfto result might appear surprising. On the other hand, since every separable Banach space is isometrically isomorphic to some closed subspace of C ([0,1], R.) [Ban], the DavieJEnfto search among the closed subspaces of C ([0,1], R.) for a separable Banach space for which there is no basis is most reasonable. Were there a basis for every closed subspace of C ([0, 1], R.), there would be a basis for every separable Banach space. There is an interesting history behind the "basis problem" for Banach spaces. Between WWI and WWII the Polish school of functional analysis flourished. Some of its members started the "Scottish book" [Mau] in which problems in analysis were proposed and rewards for some solutions were offered. Most rewards were free drinks. One prize was a live goose, for the solution of Problem 153, posed by Mazur in the following form. If fEe ([0, 1]2,R.) and f > are there in [0,1] numbers Xj,Yi, 1 :5 j :5 n,l :5 i :5 m, and is there a matrix (tij):,'j~l such that sUP(z,Y)E[0,1]2If(x,y) - Ei,j f(X,Yi)tijf(xj,y)1 < f? Mazur, loosely paraphrased, asks: When f is regarded as a "matrix" (/ZY)(Z,Y)E[0,1]2 does f have an approximate Moore-Penrose inverse f+ ~f (tij):,'j~l? (Cf. Exercise 1.3.1.4. 28.) The formulation in [Mau] of Mazur's question appears to be different from what is given above. The two are actually equivalent. The one offered here provides a useful and motivating connection with the Moore-Penrose inverse. Grothendieck [Groth] vastly extended the problem and constructed an edifice of statements, each equivalent to the affirmative solution of the problem cited above. One formulation is the following.

Jo

°

Chapter 2. Analysis

158

Let a linear transformation T be said to have finite rank iff the dimension of im(T) is finite. A Banach space V is said to have the bounded approximation property (EAP) iff a) there is a sequence {Fn}nEN of endomorphisms, each with finite rank and such that sUPn IlFn II < 00 and b) for every x in V lim IIx - Fn(x) II = 0. n ..... oo Prove that every Banach space has the BAP. Enflo proved more than the existence of a separable Banach without a basis. He constructed a separable reflexive Banach space V in which the BAP fails to hold. Since the existence of a countable basis in a Banach space implies the validity of the BAP in that space, Enflo cooked the goose. (Enflo received a live goose in a special ceremony conducted in Warsaw by some of the contributors to the Scottish book. Regrettably, other contributors, having died in the Holocaust, were absent.) [ Note 2.3.1.1: Despite the importance of the set

T~f ([O, 211") 3 () ....... cos n(},

(} .......

sin n()}nENU{O}

of trigonometric functions, it is not a basis for C (T, R.). Indeed, if Twere a basis for C (T, R.) the Fourier series for every function fin C (T, R.) would converge uniformly to f. On the other hand, there is in C (T, R.) a set E that is a dense G6 and for each f in E there is in T a set Df that is a dense G6 on which the Fourier series for f diverges [Rud, Zy]. The set Tis an (orthogonal) basis for L2 ([0, 1],C), the Hilbert space of (equivalence classes) of measurable functions 9 defined on [0,1] and such that

For a long time there were two open questions about Fourier series: i. If fEe ([0, 1], R.) does the Fourier series for

f converge anywhere? ii. If 9 E L2 ([0, 1], C) does the Fourier series for 9 converge a.e.? Many years after the questions were asked, Carleson [Ca] , in a spectacular and difficult paper, answered ii (and hence also i) affirmatively. Subsequently Hunt [Hu] extended the result to LP ([0, 1], C), p> 1.]

Section 2.3. Topological Vector Spaces

159

If B ~f {bn}nEN is a basis for a Banach space V then in the dual space V* the set B* ~f {b~}nEN of coefficient functionals such that 00

b~ : V

:3

x ~f

L

Xn b n 1-+ Xn

n=1

forms with B a biorthogonal set {B, B*}: b:'n (b n ) = omn. If V is the dual space of a Banach space W (V = W*) and if B is a basis for V the set B* can lie in W, e.g., if V is a Hilbert space or a reflexive Banach space. Furthermore if B* c W then B* is a basis for W, [Day]. However there are Banach spaces V for which B* fails to lie in W [Gel].

Example 2.3.1.1. For pin [1,00) the set

is a Banach space with respect to the norm

If p = 1 then

cn E JR, sup lenl nEN

<

oo}

and V is itself the dual of W def = Co def =

{{

dn } nEN : dn E JR, lim dn = 0 } . n ..... oo

For n in N let b n be the sequence {ank hEN such that (_l)n alk = 01k, ank = { 1

o

ifk=l if k = n ,n > 1. otherwise

Then {bn}nEN is a basis for 11 and yet bi is the sequence {(_l)n+l}nEN' which is in m \ Co: {bn }nEN is not a retrobasis for 11. Furthermore, since m is not separable, {b~ }nEN cannot be a basis for m. Every orthonormal basis 41 ~f {tPn }nEN for a Hilbert space 'It is an unconditional basis in the sense that whenever E:=1 antPn converges and IOn = ±1, n E N, then E:=lfnantPn also converges. On the other hand,

Chapter 2. Analysis

160

there are in 'It conditional bases 8 ~f {bn}nEN such that E:=l anb n converges while for some sequence {IOn = ±1 : n EN}, E:=lfnanb n does not converge. Example 2.3.1.2. If 0 < a

< 4and

Ixlo< b o = (211")t b

_I Io
2n+1 -

I-o<sin{n + l)x b 2n+2 _I - X 1 (11") 2 n = 0,1, ... then

8 ~f {bn}n=O,l, ... is a basis but not an unconditional basis for L2 ([O, 1], C) [Ge9]. Karlin [Day, Karl, Kar2] showed that in C ([O, 1], JR) no basis can be unconditional. The total variation of each function in Schauder's construction of a basis for C ([O, 1], JR) is not more than two, i.e., the Schauder system of functions is of uniformly bounded variation. Each of the Franklin functions is of bounded variation but the Franklin system is not of uniformly bounded variation. COROLLARY 2.3.1.1 to THEOREM 2.3.1.1 below provides an explanation for this phenomenon. THEOREM 2.3.1.1. LET (X,S,IL) BE A MEASURE SITUATION AND ASSUME {¢n}nEN IS AN INFINITE ORTHONORMAL SYSTEM OF FUNCTIONS DEFINED ON X. ON THE SET

E ~f {x : lim ¢n{x) ~f ¢(x) eXists} n-+oo

¢(x) = 0

A.E.

If E = 0 the conclusion follows vacuously. Since each ¢n is in L2{X, IL) it follows that C ~f U:=l {x : ¢n{x) =F O} is u-finite and ¢ = 0 on X \ C. In the argument that follows it may be assumed that E C C and that IL{E) > O. Egoroff's theorem implies that if S 3 ACE and 0 < 10 < IL{A) < 00 there is in A a subset AE such that IL (A \ A E) < 10 and ¢n ~ ¢ on AE. Since PROOF.

Section 2.3. Topological Vector Spaces

161

{ifJn} nEN is orthonormal, i.'ifJ(X)' dp,

~ i . lifJ(x) -

~ lIifJ - ifJnllco p, (A€) +

ifJn(x)1 dp, + i . lifJn(x) I dp,

v'P, (A€)

whence ifJ is integrable. Note that ~ifJn ~ lifJl on A€ and that Bessel's inequality implies limn-+co fA. sgn(ifJ(x))ifJn(x) dp, = O. Thus fA. lifJ(x)1 dp, = o and ifJ = 0 a.e. on A€. Since p, (A \ A€) can be made arbitrarily small, it follows that ifJ = 0 a.e. on A. Moreover, E is O'-finite and so there are measurable sets An of finite measure and such that E = UnEN An, whence ifJ = 0 a.e. on E.

o 2.3.1.1. IF -00 < a limsuPnENvar(ifJn) = 00.

COROLLARY

p,

= A,

THEN

< b<

00,

X

= [a, b],

S=c" AND

PROOF. If limsuPnEN var(ifJn) < 00 then, since {ifJn} nEN is orthonormal, it may be assumed that sUPnEN lifJn(a)1 < 00. The Helly selection theorem [Wi] implies the existence of a subsequence, for convenience de-

noted again {ifJn}nEN, such that limn-+ co ifJn(x) ~f ifJ(x) exists everywhere. Hence ifJ(x) = 0 a.e. and the argument used above implies the contradiction

0=

f

i[a,b]

lifJ(x) 12 dx = lim f lifJn(xW dx = 1. n-+co i[a,b]

o

For a continuous function f defined on a metric space with metric d the modulus of continuity w(j, xo, e) at a point Xo is sup {6' : d(y, xo)

< 6' =} If(y) - f(xo)1 < e} .

The uniform modulus of continuity w(j, e) is sup {6' : d(y, x)

< 6' =} If(y) - f(x)1 < e} .

Exercise 2.3.1.2. Show that if (X, d) is compact and {ifJn}nEN is, for (X, S (K) ,p,), (p,(X) < 00), an infinite, uniformly bounded, and orthonormal system in L2 (X, C)) n c (X, C) then there is a positive e such that lim sup w(ifJn , e) n-+co

= 00.

Chapter 2. Analysis

162

[Hint: Use the Ascoli-Arzela theorem.] Example 2.3.1.3. If the measure situation is ([0,1],2[0,1], JL) and

JL(E) ~£

{

#(E) 00

if E is ~nite otherWise

(JL is counting measure) then X~£ {X{:I:}}:l:E[O,l] is an orthonormal set. i. If {x{:l:n} }nEN is an infinite subsequence of X then E

~£ {x

: nl!.~X{:l:n} ~£ X eXists} = [0,1]

°

and X = on E. This example highlights the absence of any hypothesis of u-finiteness in THEOREM 2.3.1.1. 160. zz. Despite COROLLARY 2.3.1.1. 161 var (X{:I:}) = 1, x E lR. There is no contradiction because ([0,1],2[0,1], JL) is not ([0,1],.c, ,\). If {bn}nEN is a basis for a Banach space B then the biorthogonal set B ~f {bm; b:}m nEN is maximal in the sense that there is no biorthogonal ' set B that properly contains B.

,

Example 2.3.1.4. In C ([0, 1], 1R) let M be the subspace for which there is no basis (cf. the discussion, pp. 157ff., above of the Davie-Enflo example). A direct generalization of the Gram-Schmidt orthonormalization process uses the Hahn-Banach theorem to produce for M a maximal biorthogonal set that cannot be a basis.

'2.3.2. Dual spaces and reflexivity For every Banach space B there is a dual space B* consisting of the continuous linear functionals defined on B: B* ~f {x* : x*: B :3 x

1-+

C, x* is linear and continuous} .

The Hahn-Banach theorem shows that B* =1= 0. For B* there is a weakest topology 7{B*, B) with respect to which each linear functional x: B* :3 x*

1-+

x*(x) E C

is continuous. With respect to 7{B*, B), the unit ball 8 1 ~f {x* : x* E B*, IIx* II :5 1 }

is compact. The Krein-Milman theorem [Ber] implies that the 7{B*, B)closure of the convex hull of the set of extreme points of 8 1 is 8 1 itself.

Section 2.3. Topological Vector Spaces

163

Example 2.3.2.1. Assume B = C ([0,1], lR), the set of lR-valued functions continuous on [0,1]. The only extreme points of 8 1 are the constant functions J± == ±1. Indeed, if 9 is an extreme point of 8 1 assume there is in [0,1] an Xo such that Ig(xo)1 ~f 1 - 2a < 1. For some neighborhood U (xo) of xo, if x E U (xo) then -1 + a < g(x) < 1 - a. Furthermore in B there is an h such that O~h~a

=a h(x) = °if x ¢ U (xo). Thus 9 ± hE 8 1, 9 =F 9 ± h, and 9 = !(g + h) + !(g h (xo)

h), whence 9 is not an extreme point of 8 1 • Thus the closed convex hull of the set of extreme points of 8 1 is the set of constant functions of norm not exceeding one, a proper subset of 8 1 . The Krein-Milman theorem implies the following important conclusion.

The Banach space C ([0,1], lR) is not the dual space of a Banach space. Exercise 2.3.2.1. Use a modification of the argument in Example 2.3.2.1 to show that Ll ([0, 1], C) is not the dual space of a Banach space. In lRN let Co be the subset consisting of all sequences {an }nEN such that

limn--+ oo an = 0. For a ~f {an}nEN in Co let lIali oo be sUPnEN lanl. Then Co is a Banach space. Similarly c, consisting of all sequences {an}nEN such that limn--+ oo an ~f a oo exists and, endowed with the norm given to Co, is a Banach space. Exercise 2.3.2.2. Show that the in the unit ball of Co there is no extreme point. (Hence Co is also not the dual space of a Banach space.) [Remark 2.3.2.1: The Banach space Co is in fact Co (N, lR), the set of lR-valued functions defined on N, continuous in the discrete topology of N, and vanishing at infinity. More generally, if X is a locally compact and noncompact Hausdorff space then the unit ball of Co (X, lR) contains no extreme points (cf. [Ge4], Problem 83).] Example 2.3.2.2. For a ~f {an}nEN in c let T(a) be {an - aoo}nEN. Then T(c) = Co, T is continuous and linear, and ker(T) is the (onedimensional) subspace consisting of the constant sequences. The unit ball of c has just one extreme point, which gets "lost" under the action of T. Exercise 2.3.2.3. Let C in [0,1] be a Cantor set of positive measure: ,x(C) > 0. Show that if p ~ 1 then XC E LP ([0, 1], C) and that there is no continuous function in the equivalence class of Xc.

Chapter 2. Analysis

164

If B is a Banach space and B* is norm-separable then B is also normseparable [Day]. The converse is false.

Example 2.3.2.3. The Banach space [I is norm-separable and W)* is m (denoted also 1 the Banach space of all bounded sequences: (0

m ~f {

),

X

~f {Xn}nEN = ±1,

:

IIxll ~f s~Plxnl < 00 } .

EN}, an uncountable subset of m. The uncountably many open sets U(s) ~f {t : lis - til < .25}, s E S, are

Let S be { {en} nEN : en

n

pairwise disjoint. Hence no countable set can meet every U(s). Each vector x in a Banach space B may be regarded as an element f of (B*)* ~f B n :

x: B* 3 x* 1-+ x*(x) ~f f(x*) E C. Hence there is a map () : B 3 x 1-+ f E B n and 1I()(x)1I = IIxll «() is a linear isometry). The Banach space B is reflexive iff 8(B) = B**. All finite-dimensional Banach spaces and some infinite-dimensional Banach spaces are reflexive, e.g., if p > 1 then V ([0, 1], C) is reflexive.

Example 2.3.2.4. In Co let B f be the Banach subspace consisting of the set S of all sequences x ~f {Xn}nEN such that x E Co and

A direct calculation shows that if x E Bf then

IIxliB~f

IIxli Bdefined by 1

nEN,

sup Pl
['t(XP2i -1- Xp2i )2 + X~2n+ll ~ i=1

is finite and, indeed, ~ II liB :5 II IIf :5 311 liB' Since the norms II IIf and II liB are equivalent, it follows that with respect to II liB the Banach space Bf is again a Banach space BB' Since BB is not reflexive [Jal], Bf is not reflexive. The set consisting, for all n in N, of the vectors Zn

def (

=

0,

0,

... , 1, 0,

... )

T nth component

Section 2.3. Topological Vector Spaces

165

is a basis for Bf and Bj* is the set of all sequences {Fn}nEN for which n

lim II" Fnznllf n--+oo L.-J

< 00.

;=1

The map

T : Bf 3 x

1-+

(X2

-

xl. ... ,xn

-

xl. . .. )

def

= xu

is a linear isometric surjection of B f onto Bj*. Thus there is a linear isometric surjection T : B f 1-+ Bj* but the natural injection (J maps B f into a subspace W of codimension 1 in Bj*. Thus B is a Banach space that is not reflexive and that is nevertheless linearly isometric to B**. The construction is due to James. Details of the proof are to be found in [Jal, Ja2]. 2.3.3. Special subsets of Banach spaces In the set of functions f defined on JR and periodic with period 1 (f(x) == f(x + 1)) let Cp(JR,JR) be the subset consisting of the JR-valued continuous periodic functions with period 1. Each function h in Cp (JR, JR) is bounded and with respect to the norm II 1100 : Cp (JR, JR) 3 h 1-+ IIhll oo ~f sup Ih(x)1 zEJR

Cp (JR, JR) is a Banach space. Moreover, ifT is defined as JRjZ then Cp (JR, JR) may be identified with C (T, JR), or with a closed subspace of C ([0,1]' JR). Example 2.3.3.1. The set

PT~f{xl-+ i)akcoskx+bksinkx)

: nENU{O},ak,bkEJR}

k=O

of trigonometric polynomials is, owing to the Stone- WeierstrajJ or Fejer theorems, dense in C (T, JR). Furthermore PT consists of infinitely differentiable functions.

Example 2.3.3.2. The functions Gk defined in Exercise 2.1.1.15. 50 are in Cp (JR, JR), all their difference quotients are large, i.e., everywhere more than k in absolute value, and IIGkiloo $ 1. Hence if U is an open set in C (T, JR), there is in PT a p, in JR an a, and in N a Kk, depending def on p and a, and such that Hk = p + aGKk E U and everywhere Hk has difference quotients larger than k in absolute value. Let ND be the set of nowhere differentiable functions in C (T, JR) and let Ak be the set of functions having difference quotients everywhere larger

Chapter 2. Analysis

166

than k in absolute value. Then, Hk E Ak: Ak is dense in C ('][', Ii) and Ak eND. THEOREM 2.3.3.1. IN C ('][', Ii) THE SET ND OF NOWHERE DIFFERENTIABLE CONTINUOUS FUNCTIONS IS A DENSE SET OF THE SECOND CATEGORY. PROOF. For each n in N let Fn in C ('][', Ii) be the set of functions such that for some x in '][' and all positive Ihl

I

I/(x + h) - l(x)1 < Ihl - n. Owing to the compactness of '][', it follows that each Fn is a closed subset of C ('][', Ii). Since An is dense in C ('][', Ii) it follows that the interior of Fn is empty: F~ = 0 and so each Fn is nowhere dense. However SD, the set of somewhere differentiable functions in C ('][', Ii), is a subset of Un EN Fn: SDe

UFn nEN

whence SD is a set of the first category. Since C ('][', Ii) is complete and since ND = C ('][', Ii) \ SD it follows that, in C ('][', Ii), ND is a set of the second category.

o In C ('][', Ii) the set ND of nowhere differentiable functions is a dense subset of the second category and the set SD of somewhere differentiable functions is a dense subset of the first category.

Exercise 2.3.3.1. 2.3.3.1 is closed. [Hint: Assume

Show that each Fn in the proof of THEOREM

e Fn II/n - 11100 -+ 0 I/n(x n + h) - In(xn)1 <

{fn}nEN

Ihl

- n,

Ihl > 0

,n

E

IIJ

1'1.

Because '][' is compact it may be assumed that limn-+ oo Xn ~f Xoo exists. Show that for each nonzero h and each positive f

l(f(xoo

+ h) Ihl

l(xoo)1 <

- n

+ 1 f.

Section 2.3. Topological Vector Spaces

167

THEOREM 2.3.3.2. (TUKEY) IF A AND B ARE DISJOINT CONVEX SUBSETS OF A VECTOR SPACE V AND x fJ. A U B THEN THERE EXIST DISJOINT CONVEX SETS C AND D SUCH THAT Ace, BcD AND x E CUD [Tu2]. PROOF. For three vectors u, v, w in V write (uvw) iff v lies between u and w, i.e., there is in [0,1] a t such that v = tu + (1 - t)w. Note that if a1,a2 E A and bl. b2 E B then not both (xa1bt) and (xb2a2) can obtain. (Otherwise the (convex) segments a1a2 and b 1b 2 intersect in a point common to A and B.) If (xa1bt) cannot obtain let D be the convex hull of x and B and let C be A. It follows that D and A, i.e., D and C are disjoint. If (xb2a2) cannot hold let C be the convex hull of x and A and let D be B.

o Exercise 2.3.3.2. Use some version of the Axiom of Choice to conclude from THEOREM 2.3.3.2 that if A and B are disjoint convex subsets of a vector space V there exist complementary (hence disjoint) convex sets C and D such that C :::> A and D :::> B. Example 2.3.3.3. Let B be a separable, infinite-dimensional, and normed vector space. The following construction in B leads to two sequences Y ~f {Yn}nEN and Z ~f {Zn}nEN, each constituting a dense subset of B and such that Y U Z is linearly independent. Let {xn}nEN be a dense subset of B. For each n in N let Un be {x : IIx - xnll < ~ }. Define Y1 to be a nonzero vector in U1 • Because B is infinite-dimensional there is in U 1 \ span (Y1) a vector Zl. Having defined linearly independent vectors Y1, ... ,Yn, Zl. ... , Zn such that

span ({Yl. ... ,Yn}) :::> span ({ Xl.' .. ,xn}) span ({Zl,"" zn}) :::> span ({Xl. ... , x n }) and such that as well,

Yk, Zk

E Uk, 1

~

k ~ n, let Mn denote the span of

{Yk}l
Every closed proper subspace M of B is nowhere dense. (Otherwise < r} and then M contains U ~f -a+ W, a nonempty open ball centered at O. If u E B, then for a nonzero t, tu E U, whence u E M, i.e., M is not proper.) Thus, in particular, each M n , as a closed proper subspace of B, is nowhere dense and hence Un+l \ Mn =F 0. It follows that in Un+l there are vectors Yn+l. Zn+1 such that Y1, ... ,Yn+l. Zl. ... , Zn+1 are linearly independent. M contains a nonempty open ball W ~f {x : IIx - all

Chapter 2. Analysis

168

Hence each of the sets Y ~f {Yn}nEN and Z ~f {Zn}nEN is linearly independent and dense and so each of the convex hulls Conv(Y) and Conv(Z) is dense in B. Since YUZ is linearly independent, Conv(Y)nConv(Z) = 0.

Exercise 2.3.3.3. Use the results above to conclude that in any separable, normed, infinite-dimensional vector space B there are two complementary, dense, and convex subsets U and W. Exercise 2.3.3.4. Let u resp. w be a point in U resp. W above. Show that in each of the sets Su ~f {x - u : x E U} and Sw ~f {Y - w : yEW}

there are sets Yu and Zu resp. Yw and Zw like Y resp. Z above and such that u + span (Yu) n U, u + span (Zu) n U w + span (Yw ) n w, w + span (Zw) n W are four pairwise disjoint, dense, and convex subsets of B. Show that for any n in N there are at least n pairwise disjoint, dense, and convex subsets the union of which is B. Show also that B itself is the boundary of each of the n sets: In B there are arbitrarily large numbers of pairwise disjoint convex sets of which B is both the union and the common boundary.

[Note 2.3.3.1: Whereas there can be no pair of disjoint, dense, and convex subsets of lR n , for every n and every k in N there are in lRn k disjoint regions, all with the same boundary (cf. Subsection 3.1.2).] Exercise 2.3.3.5. Show that in the preceding results the hypothesis that B is separable is unnecessary. [Hint: Use Zorn's lemma.]

Exercise 2.3.3.6. Let 8 be the category of Banach spaces and continuous homomorphisms and let P(E) mean "E is separable." Show that P is a QL property: if B is a separable Banach subspace of the Banach space A and if C ~f AlB is separable then A is separable. 2.3.4. Function spaces The study of topological vector spaces began with the study of linear function spaces, e.g., C ([0,1], lR). A function space B consisting oflR-valued functions may be partially ordered with respect to the natural partial order

Section 2.3. Topological Vector Spaces

169

t: 1 t 9 iff for all x in the (common) domain of 1 and g, I(x) The same partially ordered function space B may be a lattice:

~

g(x).

I,g E B =* sup{f(x),g(x)} ~f 1 V 9 E B I,g E B =* inf{f(x),g(x)} ~f 11\ 9 E B. A function space B is an algebra iff

l,gEB=*I·gEB. Of particular interest among the function spaces are, for a measure situation (X, S, /L), the spaces V (X, C), 1 ~ p. For p in [1,00) p I def =

{..L.l

if 1 < p otherwise.

P-

00

The Holder inequality, the theorems of F. Riesz-Fischer, and M. RieszThorin imply: i. if XES and /L(X)

< 00, then 1 ~ p < q =* Lq (X, C) c

V (X, C);

ii. if cI> ~f {tPn}nEN is a complete orthonormal set in 'It ~f L2 (X, C) and 1 E 'It then 00

L 1(I, tPn) 12 < 00; n=l

if E::'=1IanI2

< 00 then there is in 'It an 1 such that

iii. (Hausdorff-Young) if X = [0, 1], if the functions in cI> are uniformly bounded, e.g., if cI> consists of the functions X

1-+

and if 1 < p

~

1, x

1-+

~ cos 21l"kx,

x

1-+

~ sin 21l"kx,

kEN,

2 then 00

1 E V (X, C) =*

L 1(I, tPn) Ipi < 00; n=l

under the same hypotheses, if E::'=llanIP < 00 there is in Vi (X,C) an 1 such that (I, tPn) = an, n E N. However the following items indicate the degree of inflexibility of the hypotheses in i and iii.

170

Chapter 2. Analysis

Example 2.3.4.1. If the measure situation is (R, C,..\) and 1 ~ p < q then neither of V (JR, C) and Lq (R, C) is a subset of the other. Indeed, since ~ < there is an a such that 0 < ap < 1 < aq. Hence If

*

I(x) ~f {xo-a g(x) ~f {xo-a

if 0 < x ~ 1 otherwise if 1 ~ x otherwise

then 1 E V (JR, C) \ Lq (R, C) and g E Lq (R, C) \ V (R, C). Thus the hypothesis J.t(X) < 00 cannot be dropped from i. The hypothesis p ~ 2 cannot be dropped from iii. The details are somewhat complicated and can be found in [Don, Zy].

Exercise 2.3.4.1. Show that if B ~f {/(x) ~fax + b : a,b,x E R} then B is a linear function space that is neither an algebra nor a lattice. Exercise 2.3.4.2. Show that if B ~f {p : p a polynomial over JR} then B is a linear function space that is an algebra but not a lattice. Exercise 2.3.4.3. Show that if B is L1 (R, R) then B is a linear function space that is a lattice but not an algebra with respect to pointwise mUltiplication of functions: I(x), g(x). Show also that B is an algebra with respect to multiplication as convolution: 1 * g(x) ~f JR I(x - t)g(t) dt. Let IIR be the set of irrational real numbers, let AIR be the set of algebraic real numbers, and let T rlR be the set of transcendental real numbers. Exercise 2.3.4.4. Show that if

1 def = XIIR g

XQ

def

= XAIR -

XTrlR

then P and g2 are Riemann integrable but (f + g)2 and Riemann integrable. Thus the set of functions 1 such that integrable is not a linear function space.

1 + g are not

12

is Riemann

Exercise 2.3.4.5. Show that if E1 resp. E2 is a nonmeasurable subset of[o, 11 resp. [2,31 and def

1 = X[O,1)UE2 g

- X[2,3)\E2

def

= X[2,3)UEl -

X[O,1)\El

then p,g2 E L1 (R) but (f + g)2 ~ L1 (R). Thus, although L2 (R,R) is a linear function space, the set of functions 1 such that 12 is Lebesgue integrable is not a linear function space.

Section 2.3. Topological Vector Spaces

171

Example 2.3.4.2. Assume that p is an integer and q is a nonzero integer and (p, q) = 1. If !

F(x)

def

=

{

q

-2 - :

-2 G(x)

H(x)

-1-!

def

= { 1+ ~

def

q

-1 -1-!

={

3 +! 3 q

q

if x = ~ and q is odd if x = .; and q is even ifxElllR if x = ~ and q is odd if x = .; and q is even ifxElllR if x = ~ and q is odd ifx=fandqiseven if x E fIR.

Then each of F, G, and H is semicontinuous everywhere whereas

F(x)

+ G(x) + H(x)

- 2+ ~ = { 2- ~

o

q

if x = ~ and q is odd if x = f and if q is even if x E fIR

and F + G + H is nowhere semicontinuous. Thus the set of semicontinuous functions in aIR is not a linear function space. Exercise 2.3.4.6. Show that if a E ][IR then both x 1-+ sin x and x 1-+ sin ax are periodic but their sum is not. Hence the set of B of continuous periodic functions in aIR is not a linear function space. Example 2.3.4.3. In the function space C ([0, 1], a) there are two possible norms: 11112 resp. II 1100. Let P resp. Q be the corresponding unit balls: p~f {I 11/112 $ I}

Q~f {I

11/1100 $ I}.

It follows that Q c P. Of greater interest is the fact that P \ Q is II 112-dense in P. If I E P and 11/1100 > 1 then I E P \ Q. If 11/1100 $ 1 let Xo be such that II (xo) I = 11/1100. It may be assumed that I (xo) ~f M > O. For a positive f there is a neighborhood U of Xo and in C ([0, 1], a) a 9 such that: i. for x in U, I(x) > Af j ii. 9 (xo) > 1 + M, 9 = 0 off U, and IIgll~ iii. i[O,l]W I/(x)1 2 dx + IIgll~ < 1.

If

h(x)
III -

hll2

{/(X) g(x)

< f.

+ I(x)

< f2j if x rt U if x E U

172

Chapter 2. Analysis

2.4. Topological Algebras

2.4.1. Derivations

If B is a commutative Banach algebra there is a corresponding (possibly empty) set Hom( B, C) \ {O} of nonzero algebraic homomorphisms of B onto C. The radical 'R of B is the set of generalized nilpotents or, equivalently, the intersection of the set of kernels of elements of Hom(B, C):

'R

= {x : =

lim sup IIxnll':

n

n ..... oo

=

o}

ker(h}.

hEHom(B,C)

In fact, limn..... oo IIxn II': always exists and is the spectral radius of x. The algebra B is semisimple iff'R = {O}. The set Cb (JR, C) of C-valued bounded continuous functions defined on JR contains the subsets C~k) (JR, C) consisting of C-valued functions with k continuous bounded derivatives. With respect to the norm

II

lI(k) :

C~k) (JR, C) 3

k

I ..... :E sup I/U)(x}1 ~f IIfIl(k) j=ozEIR

C~k) (JR, C) is a Banach algebra. Example 2.4.1.1. The set

n 00

A ~f C~oo) (JR, C) ~f

C~k) (JR, C)

k=O

in JRIR consisting of infinitely differentiable C-valued functions for which each derivative is bounded on JR is an algebra that is not a Banach algebra with respect to any norm. (Note that although no nonconstant polynomial belongs to A, it is a rather rich algebra containing, e.g., the functions In : JR 3 X ..... exp(inx), n E Z and all their finite linear combinations, the bridging functions (cf. Example 2.1.2.4. 61), etc.} Assume the algebra A is a Banach algebra with respect to some norm II lib' The Banach algebra B of continuous endomorphisms of A consists of bounded endomorphisms T : A 3 I ..... T(f} E A such that

IITII ~f sup {IIT(f}lIb

: II/lIb = I} <

00.

Section 2.4. Topological Algebras

173

The maps D : 1 1-+ D(f) ~f f' and, for g in A, Rg : 1 1-+ gl are in Band D is a derivation: D(fg) = D(f)g + 1D(g). Since DRg - RgD = Rg, and since RgRg, = Rg,Rg it follows that Rg and DRg - RgD commute: Rg (DRg - RgD) = (DRg - RgD) Rg. Generally, if T, S E Band TS - ST and S commute consider the endomorphism ~ : B 3 T 1-+ T S - ST. Then ~ is a derivation on B. Since TS - ST and S commute, ~2(T) = 0 whence by Leibniz's rule, ~k (T') = 0 if k > l. On the other hand, ~2 (T2) = 2 (~(T))2, whence by induction and Leibniz's rule it follows that ~n (Tn) = n! (~(T)t. However if M is the norm of the operator ~, then lI~n (Tn) II :5 MnllTlin whence

~(T)

is a generalized nilpotent in B. If ~ is the map D 1-+ DRg - RgD the conclusion above is that ~(D) (= R g ,) is a generalized nilpotent in B and thus that g' is a generalized nilpotent in A. Fix to in JR and let
Ln : A 3 1 1-+ I(to

+ ~2

-

I(to)

~f Ln(f).

n

Since 1 1-+ 1(to) and 1 1-+ 1(to + ~) are algebraic homomorphisms of A, they are II lib-continuous whence each Ln is II lib-continuous. Since, for all 1 in A, Ln (f) -+
whence g' = 0. Since g is an arbitrary element of A it follows that D = 0, a contradiction [SiW]. An alternative to the argument in the last paragraph stems from the characterization of the set N of generalized nilpotents of a commutative Banach algebra B as the radical 'R of B, i.e., Nis the intersection of the kernels of all algebraic homomorphisms of B into C: N = 'R = nhEHom(B,C) ker(h) [Ber, Loo]. Among the elements of Hom(A, C) are the evaluation maps E t described above. Hence the generalized nilpotent g' vanishes at each t in JR, i.e., g' = 0, and the contradiction achieved earlier is repeated.

Chapter 2. Analysis

174

[ Note 2.4.1.1: Although there is no norm with respect to which the algebra A is a Banach algebra, there is a nontrivial topology T with respect to which A is a topological algebra, i.e., each of the maps A x A:3 (x,y) 1-+ x+y E A A x A:3 (x,y) 1-+ xy CxA:3(a,x)l-+ax is continuous with respect to the product topologies in the domain and the topology Tin the range. The topology T is defined by the family

of seminorms. A neighborhood of 0 is defined by an n in N and a positive number f:

Un,E ~f {f : f

E A,

Pk(f) < f, 0 ~ k ~ n} .]

Exercise 2.4.1.1. Verify that T described above is a Hausdorff topology for A and that with respect to Tthe algebra A is a topological algebra. 2.4.2. Semisimplicity

Semisimple commutative Banach algebras are of particular interest in that they may be represented as algebras of continuous functions on locally compact Hausdorff spaces.

Example 2.4.2.1. Semisimplicity in the category of Banach algebras is a QL property, i.e., if I is a closed ideal in a Banach algebra B and if both I and B / I are semisimple then B is semisimple. [PROOF: For I a closed ideal in Band B/I ~f C, assume P(I) and P(C). The quotient norm II IIQ in C is such that for x in B,

Hence if x is a generalized nilpotent in B then x/lis a generalized nilpotent in C, whence x/I = 0 and thus x E I. Since I is semisimple it follows that

x=O.O] In connection with the notion of semisimplicity assume that

Section 2.4. Topological Algebras

175

is a short exact sequence in the category of commutative Banach algebras and continuous algebraic homomorphisms. Example 2.4.2.2. If A formula

= e(l) ([0, 11, C),

normed according to the

11/11 ~f 11/1100 + 111'1100'

°

if < p < 1, and B = {I : 1 E A, I(p) = I'(p) = o} then B is a closed ideal in A and the quotient AI B ~f e is norm-equivalent to {x

(In

1-+

a + (x - p)b : a, bEe}.

e lIa + (x - p)bll ~f lal

+ max{p, 1 -

p}lbl

and multiplication is performed "modulo x 1-+ (x_p)2," i.e., x 1-+ (x-p)2 ~f 0.) The radical 'R in e consists of all complex multiples of x 1-+ x-p whence although the original algebra A is semisimple, e is not. Semisimplicity is not preserved under quotient mappings, i.e., A and (hence) B can be semisimple while the quotient AlB is not. [Note 2.4.2.1: Malliavin showed [M] that if G is a nondiscrete locally compact abelian group, there is in the semisimple group algebra £1 (G) a closed ideal I such that L1 (G) I I is not semisimpIe. Since L1 (G) I I is semisimple iff I is the intersection of the regular maximal ideals that contain I: I = kernel (hull(I)), his argument is based on showing there is a closed ideal I such that I ~kernel (hull(I) ).1 For a commutative Banach algebra A there is the question of how extensive Hom(A, C) can be. Certainly the zero homomorphism

is always in Hom(A, C). The next example shows that nothing more than this conclusion is generally available. Example 2.4.2.3. Let A be the set H (D(O, 1)°) of functions lytic in D(O,lt ~f {z : z E

c, Izl < 1}

and continuous on D(O, 1) ~f D(O, 1)0 ~f {z

Z E

c, Izl ~

I} .

1 ana-

Chapter 2. Analysis

176 With respect to convolution I*g(z) ~f

J: I(z-s)g(s)ds as multiplication

and normed by the formula 11/11 ~ sUPlzl:9I/(z)l, A is a Banach algebra. However if I E A then a direct calculation shows

11/"*11 <- (n11111" -I)!'

n EN

(2.4.2.1)

whence 111"*11-: :5 1I/II(n - 1)!--: and Stirling's lormula implies that each I in A is a generalized nilpotent. The radical 'R of A is A itself, i.e., A is a radical algebra and Hom(A, C) = {OJ: A is the antithesis of a semisimple Banach algebra. There are nontrivial commutative Banach algebras A for which the only homomorphism of A into C is the zero homomorphism. Exercise 2.4.2.1. Show the validity of the inequality (2.4.2.1) and apply Stirling's formula to draw the conclusions stated above. [Hint: If z = re i9 , 0 :5 mathematical induction.]

r :5 1, then

II

* l(z)1 :5 11/1I2r.

Use

It is customary to confine one's attention to Hom (A, C) \ {OJ. The reasoning behind this choice is simple and revealing. By definition, an ideal I in an algebra A is a proper subalgebra such that AlUlA C I. The kernel of the zero homomorphism is A, which is not an ideal whereas the kernel of each element of Hom (A, C) \ {OJ is an ideal in A. For the commutative Banach algebra A above, Hom (A, C) \ {OJ is empty and so the radical of A, as the intersection of the kernels of the maps in the empty set, is A! A recurrent theme in the study of abstract structures is the replacement of the abstract by something more familiar, concrete, and amenable to study. For an abstract commutative Banach algebra A, the concrete paradigm is a set of continuous functions on a topological space. Each a in A is regarded as a function

a: Hom (A, C) \ {OJ 3 h 1-+ h(a) ~ a(h) E C. The set Hom (A, C) \ {OJ is endowed with the weakest topology that makes each such function continuous. As explained in [Ber, Loo], this technique leads to useful developments, e.g., the spectral theorem for normal operators in Hilbert space. If A is a radical algebra, Hom (A, C) \ {O} is empty and there is no possibility of this kind of analysis.

Section 2.5. Differential Equations

177

2.5. Differential Equations

2.5.1. Wronskians 2.5.1.1. IF THE FUNCTIONS ao(x), ... , an-l(x) ARE CONTINUOUS ON [a, b] AND IF Y ~f {Yb ... , Yn} IS A SET OF SOLUTIONS OF THE homogeneous linear differential equation THEOREM

y(n)

o

L

+

ak(x)y(k)

=0

(2.5.1.1)

k=n-l OF order n THEN Y IS LINEARLY INDEPENDENT ON [a, b] IFF THE Wron-

skian

,

Yl def

W(Y) = W

(Yl, ...

def

,Yn) = det

Yl

[

:

(n-l) Yl

IS NOT

0 [CodL].

On the other hand, if Y is not the set of solutions of a differential equation like (2.5.1.1) W(Y) may be 0 on [a,b] while Y is nevertheless linearly independent (cf. [Kr] for developments). Exercise 2.5.1.1. Show that if Yl ( X )

def

= {x2 o

if-l~x~O

Y2 ( X )

def{O

if-l~x~O

=

X2

ifO<x~1 ifO<x~1

then W (Yb Y2) = 0 on [-1, I] but Y ~f {Yb Y2} is linearly independent on [-1, I]. Show directly that Y is not the set of solutions of a second order homogeneous linear differential equation like (2.5.1.1). 2.5.2. Existence/uniqueness theorems The existence/uniqueness theorem for a differential equation 'situation of the form

Y'

= !(x, Y),

is frequently given as follows:

y(xo)

= Yo

Chapter 2. Analysis

178 THEOREM 2.5.2.1. IF

i. f IS CONTINUOUS IN A NEIGHBORHOOD OF (XO, Yo), ii. L IS LEBESGUE INTEGRABLE IN A NEIGHBORHOOD U OF xo, iii. FOR ALL Yt. Y2 IN A NEIGHBORHOOD V OF Yo AND ALL x IN U

If(x, yt} - f(x, Y2)1 :5 L(x)IYl - Y21,

(2.5.2.1)

THEN THERE IS A UNIQUE SOLUTION y, VALID IN U, OF THE SYSTEM

Y'

= f(x, Y),

Y(XO)

= Yo·

[Remark 2.5.2.1: The condition in (2.5.2.1) is a kind of generalized Lipschitz condition in which a Lebesgue integrable function L is substituted for the customary Lipschitz constant in the definition of the class Lip ct.]

= 3yi, Iyl :5 1 then

Exercise 2.5.2.1. Show that if f(x, y)

y'

= f(x, y),

=0

y(O)

has two solutions: It : x 1-+ x 3 and h : x A system of a more general form is

1-+

0, valid if

Ixl < 1.

'(Ie») -- 0 F ( X,y,y, ... ,y (2.5.2.2)

y(xo) = Yo, ... ,y(le-l)(XO) = Yle-l. The following Example 2.5.2.1, due to Rubel [Rub], shows that even if F is a polynomial function of its arguments the existence of unique solutions for (2.5.2.2) may be out of the question. Example 2.5.2.1. The differential equation

(2.5.2.3) is such that if t/J, fEe (( -00,00)) there is a solution y such that

Iy(t) - t/J(t)1 < f(t), t E (-00,00).

Section 2.5. Differential Equations

179

In fact, if 9 E aIR and

g(t)

~f { ~xp ( -

»)

(1!t 2

iftE(-l,l) otherwise

and if I' = 9 then for any constants A, B, a, f3 the function AI (at + (3) + B is a solution of (2.5.2.3). If J is a closed interval [a,b] let FJ(t) be a bridging function (cf. Exercise 2.1.2.6. 62) of the form al(at + (3) + b, and such that for a 6 in b ») and for given constants A, B, a, f3

(0, (a1

FJ(t)

={

if t E (-00, a + 6] if t E [a + 6, b - 6] if t E [b - 6, 00).

A a monotone function B

If 4> as described above is given then, on any compact interval K, 4> may be approximated by a piecewise linear function 't/J. If K is partitioned finely enough into subintervals and if bridging functions of the type FJ are pieced together to a chain that interpolates 't/J at the endpoints of the partitioning intervals, the chain approximates 't/J, hence 4>, within f(t) on K. The result of piecing together such chains over a is a solution of (2.5.2.3) that approximates 4> within f on lR. Thus the solutions of (2.5.2.3) that pass through any point (c, d) are not only not unique but can be chosen to stay within any prescribed smooth open "f(t)-channel." This phenomenon is a form of superbilurcation in that if a solution y of (2.5.2.3) is known for all x in (-00, a] the solution may be continued as a solution 01 (2.5.2.3) into any f(t)-channel containing the point with coordinates (a, y(a)). Rubel's differential equation has too many solutions. Lewy's, described next, has no solutions at all.

Example 2.5.2.2. There is in Coo (a3 ,C) an such that the partial differential equation .au

- t ax

+ au ay

-

2(

x

. )au + ty az

=

I(

x, y, z

)

I

(a smooth function)

(2.5.2.4)

has not even a weak or distribution solution on any open subset of a3 [Le]. In [Ho] there is an extended study of the phenomenon above. Hormander derives conditions that a partial differential operator P must satisfy if the equation P( u) = I is to have a solution for every smooth function I. On the other hand, the Cauchy-Kowalewski theorem implies that if the coefficients of a homogeneous partial differential equation are analytic then the equation has (local) analytic solutions.

Chapter 2. Analysis

180

Example 2.5.2.3. If P is partial differential operator and 1 is a smooth function such that P(u) = 1 has no solution then P(u) - lu = 0 is a homogeneous partial differential equation admitting no solution. Indeed, if v is a solution of P(u) - lu = 0 then lnv ~f w is a solution of P(u) = I. [ Note 2.5.2.1: The typical situation in which an ordinary differential equation, or a system of ordinary differential equations, or a partial differential equation, or a system of partial differential equations fails to have a solution is that in which the solution is required to satisfy boundary conditions, initial conditions, or smoothness conditions. The Lewy example does not involve such side conditions and yet there is no solution of the differential equation (2.5.2.4).] 2.6. Complex Variable Theory

2.6.1. Morera's theorem

The conventions D(a, r) ~f {z

H(O) ~f

Iz - al ~ r}

{I

1 is holomorphic in the region O}

are observed in this Section.

C,

THEOREM 2.6.1.1. (MORERA) IF 1 IS CONTINUOUS IN A region 0 OF IF P E 0, AND IF J8L:!. I(z) dz 0 FOR EVERY TRIANGLE bJ. CONTAINED

IN

0 \ {p} THEN

=

1 E H(O) [Rud].

Exercise 2.6.1.1. Show that if 0 ~f D(O, 1)° and

I(z)

= {~ o

if z ~ ~ otherwise

then for every triangle bJ. contained in 0 \ {O}, 1 ¢. H(O) since 1 is not continuous at 0.)

JM I(z) dz =

O. (However

2.6.2. Natural boundaries

The power series L~=o ZR ~f g(z) converges if Izl < 1 and diverges if z = 1. Hence its mdius 01 convergence is 1. On the other hand, the function I: C\ {1} 3 z 1-+ (Z-1)-1 is holomorphic in its domain and 1 = 9 in 0 ~f D(O, 1)°. Thus 9 has an analytic continuation beyond O.

Section 2.6. Complex Variable Theory

181

Exercise 2.6.2.1. Show that if 00

I(z)

= 2:zn!,

zEn ~f D(O,1t

(2.6.2.1)

n=O

then 1 E H(n) and there is no analytic continuation of 1 beyond n. (The boundary an of n is a natural boundary for I.) [Hint: ForOin1l"Qn[O,211")consider/(n~lrei/i), O~r<1, nE

N.] The sequence A ~f {an}nEN of coefficients of the series (2.6.2.1) has gaps of increasing size, e.g., if {nkhEN is the sequence of indices for which ank '" 0 then nk+l _ (k + 1)! _ k - k! - + 1. nk Indeed, according to the Hadamard gap theorem, if>.

> 1 and

the function represented by the series E~=l Ckznk cannot be continued analytically beyond the circle of convergence of the series [Rud). If the nonempty region n is not C there is in H(n) a function 1 such that an is the natural boundary for I: each point of n is a singularity of I. Indeed, if an has no limit points a Mittag-Leffler expansion provides a function having an as its set of poles. Otherwise there is in n a countable set Z such that Z' = an. The Weierstraft infinite product representation leads to an 1 in H(n) and equal to zero precisely on Z. The Identity Theorem implies that an is a natural boundary for 1 [Hil, Rud]. If the power series E::'=o anz n has a positive radius of convergence R then the series represents a function 1 in H (D(O, R)O). There is a considerable body of theorems dealing with the nature of the sequence A ~f {an} ~=o and the nature of the set S R (f) of singularities of 1 on 'fR ~f {z : Izl = R}, the boundary of the circle 01 convergence of the series. Since a limit point of singularities is a singularity, SR(f) is closed. Example 2.6.2.1. Let F be a closed subset of 'fR . •. If F is empty then for the function 1 : z t-+ z, SR(f) = F. n. If F is finite, say F = {Zl,"" ZN}, then for the function N 1 I:zt-+ 2 : - , n=l Z -

Zn

Chapter 2. Analysis

182

SR(f) = F. iii. If F is infinite, for each n in N there are in F finitely many points Pnt. ... 'Pnmn such that mn

F C k~l D

(

R)O

Pnk, n+ 1

In each D (0, Rt n D (Pnk, n!l) ° there is a Znk such that i~f IZnkl

> s~p IZn-l"I, n

= 2,3, ....

It follows that F is the set of limit points of the set

L ~f {Znk' n E N, 1 ~ k ~ m n },

i.e., F = L'. Via a WeierstraB infinite product representation there can be defined a function f holomorphic in C \ F and such that L is the set of zeros of f. In i-ii above, f is not identically zero and f is representable by a power series ~~=o anz n valid in D (0, R). In ii, iii the circle of convergence for that power series is D (0, Rt. For iii, owing to the Identity Theorem for holomorphic functions, SR(f) = F.

Example 2.6.2.2. The series ~~=o ;!,-zn! represents a function f in H (D(O, 1)°) and for which T is the natural boundary. Nevertheless the series converges uniformly in the closed disc D(O, 1).

°

Example 2.6.2.3. If < a < 1 the series ~~=o an zn 2 represents a function f in H (D(O, 1)°). The Hadamard gap theorem implies that Tis· the natural boundary for f. The series converges uniformly in the closed disc D(O, 1). Furthermore: i.

is a sequence of infinitely differentiable functions on [0,21rJ; ii. for kEN, {4>~k)}nEN converges uniformly on [0,21r]. An application of THEOREM 2.1.2.1. 53 to the sequence {4>n}nEN shows that h(IJ) ~f f (e iB ) is an infinitely differentiable function of IJ. Yet eiB is, for each real IJ, a singular point of f.

Exercise 2.6.2.2. Show that if, for k in N, lim sup In(n - 1)··· (n - k n--+oo

+ l)an l;t = 1

Section 2.6. Complex Variable Theory

183

then L~=oan2zn2 represents a function 1 in H(D(0,1)0), 8 1 (/) h(O) ~f 1 (e ill ) exists for all 0 in Ii. and is infinitely differentiable.

=

'll',

2.6.3. Square roots If 0 is a region, then 0 is simply connected iff anyone of the following obtains [Rud]: i. the region 0 is conlormally equivalent to D(O, 1)°; ii. for every 1 E H(O), if 1 '# in 0 then there is in H(O) a function h such that 1 = e h (h may be regarded as "In!"); iii. for every 1 E H(O), if 1 'lOin 0 then there is in H(O) a function g is such that 1 = g2 (g may be regarded as ".fJ").

°

(Note the elementary implication: ii => iii since e t serves for g.) Example 2.6.3.1. If 0 ~f D(O, 1)° then 1 : z t-+ z2 is holomorphic in O. Although 1(0) = yet g : z t-+ Z is holomorphic in 0 and 1 = g2. Correspondingly, although 0 \ {a} is not simply connected, nevertheless g E H (0 \ {O}) and 1 = g2.

°

2.6.4. Uniform approximation

The Weierstmfl approximation theorem is valid in the set of Ii.-valued continuous functions defined on a fixed compact interval or on a compact subset of li.n . Indeed the Stone- Weierstmfl theorem is valid in the set C (X, Ii.) of continuous Ii.-valued functions defined on a compact Hausdorff space X. The situation is quite different for C (X, C), i.e., when Ii. is replaced by C. Example 2.6.4.1. If r

'P

~f {z

t-+

°

> the set

t

akzk : ak, z

EC, n EN}

k=O

is not dense (with respect to the IllIoa-induced topology of uniform convergence) in C (D(O, 1), C). Otherwise the special function 1 : z t-+ z would be the uniform limit of a sequence of polynomials in 'P. Since 1 is not differentiable it is not holomorphic in D(O, 1)° and so 1 cannot be the uniform limit of a sequence of polynomials, since every polynomial is entire and the uniform limit of a sequence of holomorphic functions is holomorphic. Exercise 2.6.4.1. Show that if 1 E H (D(O, 1)°) nC (D(O, 1), C) then there is a sequence {Pn}nEl\I of polynomials such that Pn ~ 1 on D(O, 1). [Hint: Use Fejer's theorem and the maximum modulus theorem.]

Chapter 2. Analysis

184 2.6.5. Rouche's theorem

The statement of Rouche's theorem is an instance in which the replacement of the symbol < by the symbol $ changes a valid theorem into one that is, in the vein of Landau humor, completely invalid.

Example 2.6.5.1. The functions f : C 3 z

1-+

z2 and 9 : C 3 z

1-+

1

are such that Ig(z)1 $ If(z)1 and If(z)1 $ Ig(z)1 on C ~f {z : Izl = I}. Yet, Zh,'Y. denoting the number of zeros of the function h inside the rectifiable Jordan contour 'Y*, 2

= ZI,C '" Z/+g,C = 0

although 0= Zg,C

[Remark 2.6.5.1: valued integral

= Zg+I,C = O.

One proof of Rouche's theorem uses the Z1 211"i

1 'Y

f'(z) f(z)

+ tg'(z) d + tg(z) z,

which, if the strict inequality Ig(z)1 < If(z)1 obtains on 'Y*, exists and is a continuous, hence constant, function of t on [0,1]. If the (Rouche) condition Ig(z)1 < If(z)1 on 'Y* is replaced by Ig(z)1 $ If(z)1 the integral above might fail to exist when t = 1.] 2.6.6. Bieberbach's conjecture

Experimentation and some theoretical calculations led Bieberbach in 1916 to conjecture the next result about univalent (injective) holomorphic functions [Bi]. THEOREM (BIEBERBACH). IF f IS HOLOMORPHIC AND UNIVALENT (INJECTIVE) IN D(O,I)O AND IF, FOR z E D(O, 1)°, 00

f(z) ~f

L anz n n=l

THEN FOR ALL n IN

N,

The record of progress, before the decisive result of de Branges in 1985, in the proof of the Bieberbach conjecture is in the following list, where "19xy, Name(s), n = k" signifies that the result was confirmed in 19xy by Name(s) for the case in which n = k:

Section 2.6. Complex Variable Theory 1916, 1923, 1955, 1968, 1972,

185

L. Bieberbach, n = 2 K. Lowner, n = 3; P. R. Garabedian and M. Schiffer, n = 4; R. N. Pederson and, independently, M. Ozawa, n R. N. Pederson and M. Schiffer, n = 5.

= 6;

De Branges showed the truth of a stronger result, the Milin conjecture described below, that implies the validity of the Bieberbach conjecture. In [Br] the proof of the Bieberbach conjecture itself is given and references to proofs of the stronger results are provided. The THEOREM is sharp since if {3 E R and J is given by 00

Z 1-+

J

( ) def Z

=

Z

(1 + ei {jz)2

def~

=

L...i anz n=l

n

J is holomorphic and univalent in D(O, 1)0 and for all n in N, lanl = n. The validity of Bieberbach's conjecture is implied by the validity of the Robertson conjecture [Rob] put forth in 1936. THEOREM (ROBERTSON). IF J IS HOLOMORPHIC AND UNIVALENT IN D(O,I)O AND

then

00

J(z) = L bnz 2n - 1 , n=l THEN

Izl < 1

n

L Ibkl 2 ::; nlbl l2. k=l

In turn, the validity of Robertson's conjecture is implied by the validity of the Milin conjecture [Mi] announced in 1971. THEOREM (MILIN). IF J IS HOLOMORPHIC AND UNIVALENT IN D(O,I)O THERE IS A POWER SERIES 00

L'Ynzn n=l CONVERGENT IN D(O,I)O AND SUCH THAT J(z) = zl'(O) exp

(~'Ynzn) .

FURTHERMORE r r 1 L(r + 1 - n)nl'Ynl ::; L(r + 1 - n);;:. n=l n=l

On the other hand, if the hypothesis of univalency is dropped, the conclusion in the Bieberbach conjecture cannot be drawn.

Example 2.6.6.1. If J is z 1-+ Z + 3z 2 then univalent in D(O, 1)0 and la21 = 3 > 2 = 2lall.

J is holomorphic but

not

3.

Geometry /Topo!ogy

3.1. Euclidean Geometry

3.1.1. Axioms of Euclidean geometry

Hilbert [Hi2] reformulated Euclid's axioms for plane (and solid) geometry. Not unexpectedly, Hilbert's contribution was decisive in the subsequent study of Euclidean geometry both in the schools and in research. His axioms are grouped as follows. i. axioms relating points, lines, and planes, e.g., two points determine

ii. iii.

iv. v.

186

exactly one line, two lines determine at most one point, there exist three noncollinear points, there exist four noncoplanar points, etc.;" axioms about order or "betweenness" of points on a line; axioms about congruent ("!:!!!"): a. line segments; b. angles; c. triangles (6.ABC 5!!! .6.A' B' C' if AB 5!!! A' B', AC 5!!! A'C', and LBAC 5!!! LB'A'C', the "SAS" criterion); the axiom about parallel lines: if L is a line and if P is a point not on L then, in the plane determined by Land P, there is precisely one line L' through P and not meeting L (Euclid's "fifth postulate"). the axiom of continuity and completeness (versions of the Archimedean ordering and completeness of R.).

Section 3.1. Euclidean Geometry

187

Among the topics of research interest are those dealing with logical independence and logical consistency of axioms and theorems. Hilbert treated these problems with great thoroughness. The interested reader is urged to consult [Hi2] for all the details. Even before Hilbert's work, many questions about the axioms of geometry, in particular the parallel axiom, were resolved by Riemann's example of spherical geometry. Example 3.1.1.1. Let SI be the surface of the unit ball in JR3:

S1

=

def { ( X,y,Z ) :

x 2 + y2 + Z2

=1} .

If "line" is taken to mean "great circle" then most of the axioms of plane Euclidean geometry are not satisfied and, e.g., if Land L' are two distinct lines then they must meet (twice!): there are no parallel lines. On the other hand, Lobachevski offered a model in which all axioms of plane geometry save the parallel axiom are satisfied but in which for a line L and point P not on L more than one line passes through P and does not meet L. In Example 3.1.1.2 there is a description of Poincare's alternative model with similar properties.

Example 3.1.1.2. Let II be the interior of the unit disc in JR2 :

In II let a "line" be either a diameter of II or the intersection of II and a circle orthogonal to the circumference of II. Then it is possible to define the terms of Hilbert's system so that his axioms in i, ii, iii, v are satisfied. However if a "line" L is not a diameter of II then through the center 0 of II there are infinitely many diameters, i.e., "lines", not meeting L. A more subtle question arose in the study of Desargue's theorem illustrated in Figure 3.1.1.1 and stated next. THEOREM 3.1.1.1. (DESARGUE) WHEN CORRESPONDING SIDES OF TWO TRIANGLES IN A PLANE ARE PARALLEL, THE LINES JOINING CORRESPONDING VERTICES ARE PARALLEL OR HAVE A POINT IN COMMON (ARE "COAXIAL") [Hi2].

Despite the fact that Desargue's theorem is about triangles in a plane and refers not at all to congruence, many proofs of it depend on constructions involving the use of points outside the plane of the triangles in question and other proofs depend on the "SAS" criterion for the congruence of triangles. Moulton [Mou] showed that the proof cannot be given unless resort is made either to the axiom asserting the existence of four points that are not coplanar, i.e., to the use of solid geometry, or to the congruence axiom for triangles.

188

Chapter 3. Geometry/Topology

Figure 3.1.1.1. Desargue's theorem. any I.

ii. iii.

iv.

Example 3.1.1.3. As in Figure 3.1.1.2 below, in R. 2 let "line" mean of the following: a horizontal line; a vertical line; a line with negative slope; the union of the sides Land U of an angle having its vertex on the horizontal axis, L lying in the lower half-plane, U lying in the upper half-plane, the slopes of Land U positive, and slope of L = 2. slope of U

Section 3.1. Euclidean Geometry

189

y-axis

Figure 3.1.1.2. Moulton's plane. In the resulting model of the "plane" all the axioms save the congruence axiom for triangles are satisfied. Nevertheless the two "Desarguesian" triangles in Figure 3.1.1.2 are such that the "lines" joining corresponding vertices are neither parallel nor coaxial.

Chapter 3. Geometry/Topology

190 3.1.2. Topology of the Euclidean plane

Example 3.1.2.1. In the square having vertices at (±I, ±I) in the plane let C 1 and C2 be defined as follows: C 1 def =

{

7 (-I+t,-I+st): tE[O,I] }

u{(t,~sin(~)+~): U { (1,

C2 def =

{

~ + ~t)

tE(O,I)}

: t E [0,1] }

7 : t E [0, 1]} ( -1 + t, 1 - st)

u{(t,~sin(~)-~): U { (1, -1 + ~t)

tE(O,I)}

t E [0,1] } .

y-axis

Figure 3.1.2.1. Then C 1 and C2 are disjoint connected sets, each of which is the union of two closed arc-images and one open arc-image. Furthermore {( -1, -I)} U {(I, I)} C C 1 and {(-I, I)} U {(I, -I)} C C2 ,

Section 3.1. Euclidean Geometry

191

i.e., C 1 and C2 are two disjoint connected sets contained in a square and connecting diagonally opposed vertices, cf. Figure 3.1.2.1.

Exercise 3.1.2.1. Show that a simple arc-image or a simple open arc-image is nowhere dense in the plane. [Hint: The removal of a single point from a connected open subset of the plane does not disconnect the set.] Since an arc-image, which is a compact connected set, can be a square it is nevertheless true that there are compact connected sets that are not arc-images. Example 3.1.2.2. Let 8 be the union of the graph ofy = sin(~), 0 < x ~ 1 and the interval {O} x [-1,1]. Then 8 is compact and connected. On the other hand, regarded as a space topologized by heredity from ]R2, 8 is not locally connected, e.g., the neighborhood N that is the intersection of 8 and the open disc centered at the origin and of radius contains no connected neighborhood. Since every arc-image is locally connected [Ne], 8 is a compact connected set that is not an arc-image.

!

Exercise 3.1.2.2. Show that: i. the simple arc 8 1 defined by the parametric equations

x =t Y=

{~sint

ift~O O
is nonrectifiablej u. by contrast, the simple arc 8 2 defined by the parametric equations

x=t y

={

t2 sin! t

o

ift~O O
is rectifiablej m. the simple arc 8 3 defined by the parametric equations

x =t Y

={

t2' 1 0 sm t2'

ift~O O
is nonrectifiable but that y' exists on [0,1]. [Hint: For 8 1 and 8 3 inscribe a polygon with vertices at the origin, at a finite number of maxima, and at a finite number of minima.

Chapter 3. Geometry/Topology

192

For 8 2 show that if 0< e ~ 1 then lEI Jl + Y'(X)2 dx is bounded by a number independent of e. For 8 3 the corresponding integral is not bounded.] Example 3.1.2.3. Let 9 be the nowhere differentiable function of Exercise 2.1.1.15. 50. Let C be the simple arc defined by the parametric equations x = t, Y = g(t), t E [0,1]. Since 9 is nowhere differentiable, 9 is not of bounded variation on any interval [a, b]. Hence if ~ a < b ~ 1 then the simple arc Cab corresponding to the parameter domain [a, b] is nonrectifiable.

°

Example 3.1.2.4. Without recourse to LEMMA 2.2.1.1. 113 one can construct "by hand" an arc-image that fills the square 8 ~f [0,1] x [0,1].

6

r--I

,---

I I

I

~

!

I

~

1 1 15 1 1

11 --.., I

10 1

--..,7

sL __

I

1 __ .J1 9

121 1 1 I

I

i

l~

1

__ ---,3

.... -- _J2J I I 14

1 I

1 1 115 &.;;--

I

1 ----

1

2

3

4

5

___21 :.I

6

7

S

9

10

16 ---

11

12

13

14

15

16

Figure 3.1.2.2.

The construction parallels the procedure in Example 2.2.1.8. 115. For each partition of the interval I ~f [0,1] into 4R subintervals of equal length, partition 8 into 4R pairwise congruent squares. Order the squares linearly so that they correspond to the natural linear ordering of the partition intervals in the manner indicated in Figure 3.1.2.2. Each point t of I is the intersection of a sequence of nested partition intervals. The intersec-

Section 3.1. Euclidean Geometry

193

tion of the sequence of corresponding squares is a point P(t) of S. The map 'Y: [0,1] 3 t 1--+ P(t) ~f (x(t), y(t)) E S is continuous because as nested partition intervals descend On a t in [0,1] their corresponding squares descend on P(t) in S. The image 'Y* of the arc 'Y : t 1--+ P(t) is "space-filling." The idea is due to Hilbert. The first construction of this kind is due to Peano, after whom such arc-images are named.

°

Exercise 3.1.2.3. Show that for the arc 'Y above, if :5 a < b :5 1 the corresponding arc-image 'Y:b is nonrectifiable. [Hint: The arc-image in R? of a rectifiable arc is a two-dimensional null set (A2)'] Example 3.1.2.5. The following construction produces in RR a dense set A that meets every vertical and every horizontal line of R2 in exactly One point, i.e., A is the dense graph of a bijection f : R 1--+ R. The function f is defined on a countable dense set D by induction and then on R according to the formula f(x)

= { f(x) x

I?

if x E otherWise.

Let S ~f {(xn,Yn)}nEN be an enumeration of Q2, let BI be the square [-1,1)2 in R?, and let (xn,Yn) be the first term in SnB I . Then f (xn) ~f Yn. Divide B2 ~f [_2,2)2 into 22 congruent subsquares B 2l , .. . , B 222. Let (X21' Y21) be the first term in S n B21 and such that X2l '" Xn, Y21 '" Yn. def Then f ( X21 ) = Y21· ... Let (X222 , Y222) be the first term in S n B222 and such that X222 is nOne of the Xij and Y222 is nOne of the Yij previously chosen for defining values of f. Then f (X222) ~f Y222. Divide Bn ~f [-n, n)2 into 2n congruent subsquares B nl , ... , B n2 n. Choose (Xnl' Ynl) as the first term of S n Bnl and such that Xnl is none of the Xij and Ynl is none of the Yij previously chosen for defining values of f. Then f (Xnl) ~f Ynl . ... Choose (Xn2n, Yn2n) as the first term of S n Bn2n and such that Xn2n is nOne of the Xij and Yn2n is nOne of the Yij previously chosen for defining values of f. Then f (Xn2n) ~f Yn2n . ... Exercise 3.1.2.4. Show that the procedure described above yields: a) a set D dense in Rj b) a set { (x, f(x)) : XED} dense in R2j c) a bijective function f for which the graph {(x,f(x)) : x E R} is dense in R2. Example 3.1.2.6. Knaster and Kuratowski constructed in R2 a Connected set S containing a point P such that S \ {P} is, like the Cantor set Co, totally disconnected [KnKu].

Chapter 3. Geometry/Topology

194

In Co there is the set X of endpoints of intervals deleted in the construction of Co and there is the complementary set Y ~f Co \ X. Let P be the point (0.5,0.5) in lR.2 and for each point c in Co let L(c) be the line segment joining P to c. For each x in X let Rz in L(x) be the set of points having rational ordinates. For each y in Y let Ry in L(y) be the set of points having irrational ordinates. Then, as the following argument reveals, the properties described in the preceding paragraph obtain for S ~f UzEx Rz U U EY Ry. (Note that Y n S = 0.) To show that S \ {P} is totally disconnected note that for every c in Co, S n L(c) is (totally) disconnected. Hence if Z is a nontrivial connected subset of S\ {P} then Z meets two different line segments L(ct} and L(C2)' Thus there is through P a line L that meets [0,1] \ Co between Cl and C2. If Z is connected then Z meets L, hence only at P, which is impossible since P rt Z. Thus S \ {P} is totally disconnected. The proof, due to Knaster and Kuratowski, that S is connected is offered in the format of a sequence of pairs Statement(Reason). Assume S is the union of two sets A and B such that PEA and (A n B) U (A n B) = 0. It is shown below that B = 0, which proves that S is connected. For each c in Co let l(c) be the lower endpoint of the longest line segment Ac contained in L(c), containing P, and not meeting B. 1. For all c in Co, l(c) E A and l(c) rt B. (All the points on AcnRc are in A and A n B = 0.) 2. If l(c) rt Co then l(c) rt s. (If l(c) ¥= c then l(c) E B whence l(c) rt A. See 1.) 3. The ordinate of each l(y) is rational. (See 2. and the definition of L(y).) Let E be the set ofalll(y), y E Y and let {rn}nEN be an enumeration of (0,0.5]nQ. Then, Eo denoting En(x-axis) and En denoting the intersection of E with the horizontal line through (0, r n ), E = Eo U U:=o En. 4. Each of the following relations is valid. EoCY En C AnB, n E N En C AnB, n E N En

nS

= 0,

n E N.

(See the definitions and 1.) 5. If n E N and x E X then En n L(x) = 0. (If Z E En n L(x) then the ordinate of Z is rational whence Z E S. Since the ordinate of Z is not zero, Z rt s. See 2.) Let Yn be {c : c E Co, En n L(c) ¥= 0 }. 6. For n in N, Yn C Y C Eo U UnEN Yn• (See 5.) 7. Co = X U Eo U UnEN Yn• (See 4.)

Section 3.1. Euclidean Geometry

195

8. Each Y n is compact. (Each Yn is the central projection from

(~,~) of the compact set En onto the x-axis.)

9. Each set Yn is nowhere dense in Co (The interior (relative to Co) ofYn is empty since every point ofY is a limit point of X and Yn C Y, see 6. Furthermore Y n is closed (8) and so Y,:' = (Yn)o.) 10. The set Eo is dense in Co. (The set Z ~f XU UnEN Y n is of the first category in Co. The set Co is a complete metric space, and Eo = Co \ Z [Kur].) 11. If e E Eo and z E S n L( e) then z E A. (There is in Yay such that e = l(y) (= l(e)), see 1.) 12. The closure of A contains S: A :::> S, i.e., A :::> AUB, whence AnB (= 0) = B. (Because Eo is dense in Co it follows that snUeEEo L(e) is dense in S.)

o The work of Brouwer led to a method for defining in 1R? three regions RbR2,R3 having a common boundary: o~ = oRj , 1 :5 i,j :5 3, [Bro]. An appealing description of his construction can be given in the form of a algorithm for revising the topography of a rectangular and sea-bound island in which there are two square lakes, one consisting of warm fresh water and the other consisting of cold fresh water. For each n in I'll three canals are dug successively so that: i. first, a canal brings water from the sea within a distance of ~ from

every point of dry land on the island; ii. second, a canal brings water from the lake containing warm fresh water within a distance of ~ from every point of the dry land remaining on the island after the construction of the canal for sea water; iii. third, a canal brings water from the lake containing cold fresh water within a distance of ~ from every point of dry land remaining on the island after the construction of canal for sea water and the canal for warm fresh water. What is left of the dry land on the island when the cycle i-iii has been carried out for all n in I'll, e.g., in a "one-year plan" in which cycle 1 is completed in half a year, cycle 2 in a quarter of a year, ... , cycle n in the 2- n th part of a year, is the common boundary of the newly created regions Rl resp. R2 resp. R3 consisting of of sea water resp. warm fresh water resp. cold fresh water. The description above is vivid but does not deal with, e.g., the problem of building the canal for the sea water so that the lakes are thereby not separated from each other. In what follows, a systematic formulation of a more general construction is offered. In Figure 3.1.2.3 there are square regions R l , ... , Rn inside a fixed closed rectangular region K.

Chapter 3. Geometry/Topology

196

K

••• IoIR

L:J

Figure 3.1.2.3.

Figure 3.1.2.4.

In Figure 3.1.2.4 the region Rl is "exploded" by an auteomorphism
Section 3.1. Euclidean Geometry

197

set that is the union of ~ together with a compact f-pad covering oR;, i ¥- k, l/Jk,e (Rk) c K \ Ri,e

U

i¢k

l/Jk,e (~,e) =

~,e, i

¥-

k,

whence every point in the compact set Kk,e ~f K \ farther then f from l/Jk,e (Rk); iii. K \ l/Jk,e (Rk) is connected.

Ui¢k

~ is not

[Remark 3.1.2.1: The image of Rk under the map l/Jk,e occupies most of the complement in K of the union of other sets ~,e, i.e., for i different from k:

Let

be the homeomorphism that is the composition of the maps C}}. ~f l/Jn,e 0 ••• 0 l/Jl,e' Then, according to Brouwer's invariance of domain theorem, each c}}e (~) is a region and thus every point of the compact set c}}e

l/Ji,., 1 :::;

i :::; n:

is not farther than f from each of the regions more, if f' < f then

Hence, if

F~f

C}}. (~),

1:::; i :::; n. Further-

n

Kt

kEN

'Ri ~f

U

c}}

t (Ri), 1:::; i :::; n,

kEN

then each point of the nonempty compact set F is at distance zero from each region 'R i , i.e., F is the common boundary of the n disjoint and intertwined regions 'R i , 1:::; i :::; n. For n in N, there are in ]R2 n pairwise disjoint regions

Chapter 3. Geometry/Topology

198

having a compact set F as their common boundary, i.e., if P E F then every neighborhood of P meets each region and its complement.

[ Note 3.1.2.1: Appel, Haken, and Koch [ApHI-ApH4, ApHK] positively resolved the four-color problem. When n > 4 the construction above negatively resolves the four-color problem if it is stated loosely, viz.: Can any map in JR2 be colored with four or fewer colors?] Example 3.1.2.7. Each of the regions 'Ric is a non-Jordan region since the complement of the union of a Jordan region and its boundary is precisely one region. Nevertheless each 'Ric is the interior of the closure 'Ric of 'Ric: 'Ric = ('Ric) 0. Only the inclusion 'Ric ::) ('Rlct needs proof. However, 'Ric = 'Ric U 0'R1c and K \ ('Ric) is the union of the other 'Ri. Hence, if P E 0'R1c and P is in (Rk) ° then some neighborhood of P fails to meet each of the other 'Ri , in contradiction of the fact that 0'R1c is also O'Ri , i.e., 'Ric ::) (Rk) 0. Example 3.1.2.8. In contrast to the non-Jordan regions 'Ric above, the non-Jordan region R ~f

{

(x, Y) : x 2 + y2 < I} \ { (x, 0) : 0::; x < I},

consisting of a circular region from which a slit is deleted, is not the interior of its closure: = {(x, y) : x 2 + y2 < I} i:- R.

CRt

In [KnKu] there is a wealth of other results about pathologies of the Euclidean plane. 3.2. Topological Spaces

3.2.1. Metric spaces

Exercise 3.2.1.1. Show that ifJR is metrized according to the formula: d( X,y ) ~ -

Ix - yl 1 + Ix-yl

then for n in N, each set Fn ~f [n,oo) is closed and bounded and yet nnENFn

= 0.

[ Note 3.2.1.1: Although each set Fn is bounded and closed, it is not compact in the d-induced topology for JR.]

Section 3.2. Topological Spaces

199

Exercise 3.2.1.2. Show that if N is metrized according to the formula

1m - nl d( m,n ) ~f - '---"':' mn

then {n }nEN is a nonconvergent Cauchy sequence.

[ Note 3.2.1.2: Metrized in the manner just described N is not Cauchy complete while in the metric inherited from JR, N is Cauchy complete. In each metric N is homeomorphic to itself in the other metric. Hence Cauchy completeness is not a topological invariant.] In a complete metric space a decreasing sequence

of balls with diameters tending to 0 has a nonempty intersection:

If the sequence of balls merely decreases, i.e., if Bn C Bn+l, n E N, but the diameters do not tend to 0, the intersection can be empty:

Exercise 3.2.1.3. Show that if N is metrized according to the formula d(m,n)

~f

{Io+ m~n

~fm # n

Ifm

=n

then: i. N is a complete metric space; ii. each set Bn ~f {m : d(m,n) ~ 1 + (2~)}' n E N, is a closed ball; iii. Bn :::> Bn+l, n E N; iv. nnEN Bn = 0. In a normed vector space the closure U of an open ball:

U ~f {x :

IIx - all < r}

is the corresponding closed ball:

B ~f {x

IIx - all ~ r } .

Chapter 3. Geometry/Topology

200

Furthermore, if a vector space contains closed balls Bl resp. B2 of radii rl resp. r2 and if Bl C B2 then rl < r2. In a metric space where the metric is not derived from a norm such relations can be different. Exercise 3.2.1.4. Assume X is a set and #(X) according to the formula d(x,y) ~f

{Io

~

2. Metrize X

if x =1= ~ otherwise.

For a fixed x in X let U resp. B be the open resp. closed ball centered at x and ofradius 1. Then {x} = U c B = X and U = U =1= B. Exercise 3.2.1.5. Let X be the closed disk { (x, y) : x 2 + y2 ~ 9 } in the metric inherited from 1R2 • Show that if B2 ~f X and

then Bl C B2 while their respective radii are 4 and 3. 3.2.2. General topological spaces

In metric spaces, a) sequences have unique limits, b) derived sets are closed, c) the Bolzano- Weierstrap theorem obtains, etc. In the looser domain of nonmetrizable spaces there are correspondingly less intuitive phenomena. The next discussion provides only a very small sample of the richness of topopathology. In [88] there is a far mOre extensive treatment of the subject. In what follows, 0 denotes the set of open sets of the topology of a set X. For the trivial topology is 0 = {0,X} and for the discrete topology 0, is P(X) ~f 2x , the power set consisting of all subsets of X. When Y is a subset of a topological space X, the derived set (the set of limit points of Y) is denoted Y'. Let 'Rn denote the usual topology for IRn : a countable base for 'R2 is the set of all open disks { (x, y) : (x - a)2 + (y - b)2 < r2, a, b, r E Q}.

o

Example 3.2.2.1. Let X be a space with the trivial topology. i. If #(X) ~ 2, y E X, and Y ~f {y} then Y' = X \ {y} and hence Y is not closed. ii. If#(X) ~ 2 andN~f {X~hEA is a net (in particular ifNis a sequence) then every point in X is a limit of N, even if the net is a constant, i.e., if there is in X a point y such that for all A, x~ = y.

Exercise 3.2.2.1. Let X be IR in which, by abuse of language, 0 = 'R1 n Q. Show:

Section 3.2. Topological Spaces

201

z. Q is a countable dense subset of X j n. X \ Q ~f KJR (the set of irrational real numbers) inherits from 0 the discrete topologyj iii. there is in KJR no countable dense subset. Exercise 3.2.2.2. Let X be the closed upper half-plane

{{x,y) : x,yEJR, y>O}l:J{(x,y) : xEJR,y=O}~f Al:JB. Let a set U be in the base for the topology 0 of X iff U is an open subset of A or U is of the form { (x, y) : {x - a)2 + {y - b)2 < b2, b> O} l:J{(a, On. Show that the (countable) set of all points with rational coordinates is dense in X but that B (= JR) inherits from X the discrete topology and thus contains no countable dense subset. Exercise 3.2.2.3. Show that the spaces X in Exercises 3.2.2.1, 3.2.2.2 are not separable,i.e., that neither contains a countable base for its topology. The topology of a space can be specified by the set of all convergent nets. On the other hand, the set of all convergent sequences can fail to determine the topology of a nonmetrizable space. Exercise 3.2.2.4. Assume #(X) > #(N). Let 0 consist of 0 and the complements of all sets 8 such that #(8) ~ #(N). Show that: i. the sequence {Xn}nEN converges iff Xn is ultimately constant, i.e., iff there is in X an x and there is in N an m such that Xn = x if n > mj

n. 0 is strictly weaker than the discrete topology D and, in D, a net N' converges iff it is ultimately constantj iii. if A is an uncountable proper subset of X and y E X \ A then y is a limit point of A and yet no subsequence of A converges to Yj iv. if A is a proper subset of X and y E X \ A there is a net {a~} ~EA contained in A and converging to y.

[Hint: For iv let A be the set of all neighborhoods of y and partially order A by inclusion: A >- A' iff A cA'. For each A in A let a~ be a point in A n A.] If 1 ~ P < 00, for lP there are the norm-induced topology N derived from the metric d{a, b) ~f lIa - blip and the weak or a (IP, (lP)*) topology W for which a typical neighborhood of 0 is

Chapter 3. Geometry/Topology

202

Exercise 3.2.2.5. Show that in lP every weak neighborhood of 0 contains a norm-induced neighborhood of 0, but that every weak neighborhood of 0 is norm-unbounded. (Hence N is strictly stronger than W and every N-convergent sequence is W-convergent.) Exercise 3.2.2.6. Show that in 11 every weakly convergent sequence is norm-convergent. (Hence, although W is strictly weaker than N the sets of convergent sequences for the two topologies are the same.) [Hint: Assume that for some positive D, all the terms of a sequence S converging weakly to 0 have norms not less than D. Let x(n) ~f (x~n), ... , x~), ... ) be the nth term of S. Then, d m denoting the sequence {Dmn}nEN (an element of 11*), it follows that for m in N, d m (x(n») = x~) - 0 as n - 00. Let nl be 1 and let ml be such that

E:=ml+lIX~dl < ~. There is an

~ml I (n 2 )1 LJm=1 xm

6 <~.

L et

m2

n2 such that

be such t hat LJm=m2+1 ~oo IXm(n2 )1 <~, 6

whence E::~ml+lIX~2)1 > ~. In this manner there are definable strictly increasing sequences {nl! n2, . .. } and {ml! m2, . .. } such that

If

then a ~f {am} mEN is in 100 = 11 * . For n in the sequence {n2,n3, ... }, a(xn ) ~ ~ and thus an infinite subsequence of S lies outside the weak neighborhood Ua,t' a contradiction.] [Remark 3.2.2.1: The construction above uses the "moving hump" technique, the quantities

serving as "humps."] Exercise 3.2.2.7. Show that if

Section 3.2. Topological Spaces

203

then 0 is a weak limit point of S ~f {y(n,m) }m,nEN and that no subsequence of S converges weakly to O. [Remark 3.2.2.2: Despite the topopathologies exhibited above, the following should be observed. If E is a Banach space then Bi, the unit ball of the dual space E" is compact in the weak" or 0' (E", E) topology for E". ( A typical weak" neighborhood of 0 in E" is, for n in N and Xl!"" Xn in E, a set of the form UXl, ... 'Xn;E~f{x" :

IX"(Xi)l
If E is separable then Bi in the weak" topology satisfies the first axiom of count ability.] A map! : X 1-+ Y between topological spaces is continuous, open, or closed iff, correspondingly, !-l(open set) is open, !(open set) is open, or !(closed set) is closed.

Exercise 3.2.2.8. Show that! : neither open nor closed.

]R 3 x

1-+

eX cos x is continuous but

Exercise 3.2.2.9. Show that if X ~f {(cos(},sin(}) : 0~(}<211'} then:

z. X 3 (cos(},sin(}) 1-+ (} E [0,211') is both open and closed but is not continuous (at (1,0)); n. . ) def { if ~ ~ 11' 9 : X 3 ( cos (}, sm (} = x 1-+ (} _ 11' if 11' < (} < 211'

°

° (}

is closed but neither continuous nor open.

[Hint: For ii note that Discont(g) = {(I,O)} ~f {Pl. Reduce the problem to: Xn -+ P and 9 (xn) -f+g(P) and a contradiction.] Exercise 3.2.2.10. Show that ]R2 3

(x, y)

1-+ X

E ]R

is continuous and open but not closed. Exercise 3.2.2.11. Show that H in Example 2.1.1.3. 51 is open but neither closed nor continuous. [Hint: If n < Xn < n + 1 and H (xn) E (n~l' ~) for n in N then X ~f {Xn}nEN is closed and H(X) is not.]

Exercise 3.2.2.12. Show that [0, 2] 3 x

1-+ {

~

_

1

if x E [0,1] if x E (1,2]

Chapter 3. Geometry/Topology

204 is continuous and closed but not open. Example 3.2.2.2. Assume

X~f

00

U ((3n,3n+l)U{3n+2}),

y~f(X\{2})U{I}

n=O

f:X3X~{~ :!:~~eY g:Y3y~

{ ~~-1

ify~1

if3
y- 3

Then both f and 9 are continuous bijections but neither is a homeomorphism since in any homeomorphism h: Y ~ X, h((O,I)) must be some interval (3n,3n + 1) and then h(l) cannot be defined. In Figure 3.2.2.1 there are depicted two subsets X and Y of ]R2. Each is a continuous bijective image of the other, although X and Yare not homeomorphic. Of course, although the maps Sand T described below are continuous bijections, one from X onto Y and the other from Y onto X, they are not inverses of each other.

0001 I I

x

000101

y

p

R

Figure 3.2.2.1.

Q

Section 3.2. Topological Spaces

205

The vertical segments are of length 2 and are shorn of their upper endpoints. The circles are of diameter 2. The continuous bijection S : X

1-+

Y is defined as follows:

S : horizontal line 1-+ horizontal line S: Ai 1-+ C 1 S: A21-+ Bb (S: [0,2) 3 t 1-+ (cos 1rt,sin 1rt)) S: An+2 1-+ B n+b n E N S: Dn 1-+ Cn+bn E N. The continuous bijection T : Y

1-+

X is defined as follows:

T : horizontal line 1-+ horizontal line T : Bl 1-+ Dl T : Bn+! 1-+ An, n E N T: C 1 1-+ D2 (T: [0,2) 3 t 1-+ (COs1rt,sin1rt)) T : Cn +! 1-+ D n +2, n E N. If H : Y 1-+ X is a homeomorphism, H (B 1 ) and H (C2 ) must be circles and Dh +1 and the segment PQ must map into a corresponding segment P'Q' in X. Removal of R from Y leaves a set with three components. Removal of H(R) from X leaves a set with two components. Hence X and Y are not homeomorphic. In IR3 there are many homeomorphic images of Sl ~f {x : IIxll = 1 }, the surface of the unit ball centered at the origin. For example, the octahedron or double pyramid Dil

P2 ~f { (x, y, z) :

Ixl + Iyl + Izl = 1 },

is homeomorphic (via central projection) to Sl and is regarded as tamely embedded or tame, because the complement of P2 is simply connected: every simple closed curve 'Y : [0,1] 3 t 1-+ IR3 \ P2 in IR3 is homotopic (in IR3 \ P2 ) to a point. By contrast, the complement of the surface of a torus T swept out in IR3 by a unit circle orthogonal to the xy-plane and centered on the circle

{ (x, y, z) : x 2 + y2

= 4, z = o}

is such that the circle

C 1 ~f

{

(x, y, z) : (y - 2)2

+ z2 =

5, x =

°}

is not homotopic to a point in IR3 \ T. The complement of T is not simply connected.

206

Chapter 3. Geometry/Topology

Alexander [AI] produced a surface E homeomorphic to 8 1 and yet such that R,3 \ E is not simply connected. The surface E is a wildly embedded or wild sphere. The surface depicted in Figure 3.2.2.2 is a version of E, Alexander's homed sphere, on which the set of "horn-tips," a homeomorphic image of the Cantor set, constitutes a set of "obstacles to homotopy."

Figure 3.2.2.2. Alexander's horned sphere.

Subsequently Artin and Fox [FoAr] developed the theory of wild em-

Section 3.2. Topological Spaces

207

beddings. They gave an example of a simple arc 'Y : [0,1] 3 t 1-+ IR3 such that 'Y. is snarled and (wild) cf. Figure 3.2.2.3. The endpoints 'Y(O) and 'Y(I) are obstacles to homotopy because there is a closed curve", : [0, 1]1-+ IR3 \ 'Y. that is not homotopic to a point in IR3 \ 'Y •. The arc-image 'Y. can be used to form a wild sphere as follows. Let a long hose be tapered to a point at each end and then snarled so that the axis of the hose lies along 'Y.. The surface E of the resulting snarled hose is homeomorphic to the surface 8 1 . Nevertheless E is a wild sphere since the points that are the two pinched ends of the hose are obstacles to homotopy in the unbounded component of IR3 \ E.

There exist wild spheres in IR3.

Figure 3.2.2.3. The Artin-Fox arc-image 'Y•. In Figure 3.2.2.4 there is an attempt to indicate the early stages of the construction of Antoine's necklace N. For n in N, at the nth stage there are 2n linked solid tori, the union of which is a compact set Kn. The sequence of the Kn is nested: Kn+l C Kn, and nnEN Kn = N.

Figure 3.2.2.4. Antoine's necklace. Antoine's necklace is homeomorphic to the Cantor set Co, i.e., N is a nowhere dense, perfect, and totally disconnected set, and yet IR3 \ N is not

208

Chapter 3. Geometry/Topology

simply connected, e.g., the circle lying on the surface of the first torus is not homotopic to a point in ]R3 \.N. 3.3. Exotica in Differential Topology

For homeomorphic topological spaces X and Y on each of which there is a differential geometric structure there arises the delicate question: Are X and Y diffeomorphic, i.e., is there a Coo homeomorphism, a diffeomorphism, I : X 1-+ Y? Such questions aroused great interest when it was discovered by Brieskorn, Hirzebruch, and Milnor [Bri, Milnt] that the sphere 8 7 is susceptible of several differential geometric structures no two of which are diffeomorphic. The surfaces are rather easily described, although the absence of diffeomorphisms between them requires more discussion than can be given here. Example 3.3.1. When N 3 k

Xk (ao, .. . , ak)

8 2k +1

~f

{

~f

{

~

3, N 3 aj

(zo, . .. ,Zk) : Zj E

(zo, ... ,Zk) : Zj E e, 0

e,

0

~ j ~ k,

~

2 there are defined:

~ j ~ k,

t

t

zji

=

o}

.1=0

ZjZj

=1}

.1=0

W 2k - 1 (aO, ... , ak ) def = X k (aO,"" ak ) n 8 2k +1 . Thus Xk is a set of complex dimension k, 8 2k +1 is a set of complex dimension k, and W 2k - 1 (ao, ... , ak) is a set of complex dimension k -1 in ek+1. Viewed in ]R2k+2 the sets in question have real dimensions 2k + 1, 2k + 1, and 2k - 1. Then W 7 (3, 6r - 1,2,2,2), 1 ~ r ~ 28, are all homeomorphic to 8 7 and are pairwise nondiffeomorphic. Furthermore, any oriented (7dimensional) differentiable manifold homeomorphic to 8 7 is diffeomorphic, for some natural number r in [0,28], to W 7 (3, 6r - 1,2,2,2). In [Law] there is an exposition of and a large bibliography about exotica, i.e., homeomorphic but nondiffeomorphic differential geometric structures. In particular, there is an extensive discussion of the following phenomenon, among many others. There are (uncountably many) nondiffeomorphic differential geometric structures for ]R4. On the other hand, if 4 ~ kEN then any two differential geometric structures for ]Rk are diffeomorphic. Related to the developments just described is the almost complete resolution of the Poincare conjecture, relating homotopy and homeomorphism of differential geometric structures.

Section 3.3. Exotica in Differential Topology

209

Two arcs "Ii : [0,1] 3 t 1-+ X, i = 1,2, are homotopic iff there is a continuous map h: [0,1]2 3 (s, t) 1-+ X such that h(O, t)

= "Il(t) and h(l, t) = "I2(t).

Arcs are special instances of continuous maps (between the topological space [0,1] and a topological space X). There is a more general notion of homotopy of arbitrary maps fi : X 1-+ Y, i = 1,2, between arbitrary topological spaces X and Y. Two such maps are homotopic iff there is a continuous map h: [0,1] x X 1-+ Y such that for all x in X, h(O,x)

= It(x) and h(l,x) = h(x)

in which case one writes It e:! h. Two topological spaces X and Yare homotopically equivalent iff there are continuous maps

f :X

1-+

Y and 9 : Y

1-+

X

such that the composition maps fog: Y

1-+

Y resp. 9 0 f : X

1-+

X

are homotopic to the identity maps

ly : Y 3 Y 1-+ Y E Y resp. Ix : X 3 x

1-+

X E

X.

Poincare's conjecture in modern terms asserts that an n-dimensional manifold homotopically equivalent to the sphere

sn ~f {x

: x

E ]Rn,

IIxll = 1 }

is actually homeomorphic to sn. In the early part of the twentieth century the conjecture was known to be valid if n = 1 or n = 2. In 1961 Smale [8ml] showed the conjecture is valid if n > 4 and in 1982 Freedman [Fr] showed it is valid if n = 4. At this writing, the resolution of the conjecture for n = 3 has not been announced. [ Remark 3.3.1: In [Fr] the existence, noted earlier, of nondiffeomorphic and yet homeomorphic images of ]R4, falls out as a by-product of the general thrust of the work.]

4.

Probability Theory

4.1. Independence

The theory of probability deals with a probabilistic measure situation (X, S, P), i.e, a measure situation specialized by the assumptions a) XES and b) P(X) 1. Elements of S are events and measurable functions in R X are called random variables. If I is an integrable random variable then I(x) dP. its expected value or expectation is E(f) ~f According to Kolmogorov, the founder of the modern theory of probability, the distinguishing features of the subject are the notions of independence of events and independence of random variables [Ko!, K02].

=

Ix

A set E ~f {A~} ~EA of events A~ is said to be independent iff for every subset {An : Ai:f:. Aj if i :f:. j, 1 ~ n ~ N} of E,

A set F ~f {f~} ~EA of random variables is said to be independent iff for every finite subset {In : li:f:. /;' if i :f:. j, 1 ~ n ~ N} of F and every finite set {Bl, ... ,BN } of Borel subsets of R, {fl1(Bl), ... ,fNl(BN)} is independent. One says that the events belonging to a set E are themselves independent if E is; similarly one says that the random variables belonging to a set F are themselves independent if F is. 210

Section 4.1. Independence

211

Exercise 4.1.1.

z. Show that if E and F are independent events then

u. Show that if It, ... , In are independent random variables, if {Pi}f=l C N, and if If', is integrable, 1 :5 i :5 n, then

1IT

If' (x) dP =

X i=l

IT 1Ir

(4.1.1)

(x) dP.

i=l X

m. Show that if, in the definition of independent random variables, the set {/11(B1), ... , IN1(BN)} is independent whenever the Borel sets B i , 1 :5 i :5 N, are open intervals then the random variables are independent. For any event A, {0, X, A} is independent and for any random variable and any constant function c, {c, f} is independent. These are the trivial instances of independence. Note also that if is an independent set of events and if some events in are replaced by their complements to produce a new set then is also independent. A discussion motivating the definitions of independence of events and of random variables is given in [Halm]. A significant fraction of classical probability theory concerns itself with theorems about sets of independent random variables [Loe]. In the remainder of this Section there is an attempt to reveal the "reluctance" of sets of events resp. sets of random variables to be independent. (Salomon Bochner said that since (4.1.1) is "repugnant" to a set of functions, he was not surprised at the singularity and hence the importance of independence as a phenomenon. )

I

e,

e

e

e

Exercise 4.1.2. Let P in the measure situation (N, 2N, P) be the discrete measure:

P(n) ~f

{

1- E:'2 2- n !

2- n !

if n = 1 if n ~ 2.

Show: z. if 1 < m, n E N then there is in N no k such that k! = m! + n!j u. trivial instances aside, there are no independent sets of events in 2N j iii. trivial instances aside, there are no independent sets of random variables defined on N endowed with the a-algebra 2N and the measure

P.

212

Chapter 4. Probability Theory [Hint: If e is an independent set of events then any two-element subset of e is independent.]

Example 4.1.1. If

} X def = {b a, ,c, d,

Sdef =

2X , P (a) = ... = P (d) =

4"1

then any two of the events in e ~ {{a, b}, { a, c}, {b, c}} are independent but e is not independent. Furthermore if P(a) = 0.1, P(b) = 0.2, P(c) = 0.3, P(d) = 0.4 then two events are independent iff one of them is X or 0: only trivial instances of independence are at hand. Events as such are not independent. The independence of events is completely determined only with respect to the probability measure P. A large class of probabilistic measure situations consists of those that are measure-theoretically equivalent to S ~f ([0, I], C,.\)

[HalmJ. Hence, counterexamples for S have a large range of relevance. In the following discussion it is assumed that the probabilistic measure situation is S. Two Borel sets At and A2 are regarded as equal if their symmetric difference

is a null set. Two random variables I and g are regarded as equal if I -g = 0 a.e. This kind of "equality" is equality modulo null sets. For a set S of random variables let Ind(S) denote the set of random variables I such that S U {f} is independent. In particular let Const denote the set of constant functions. Hence for any set S of random variables, Const C Ind(S). Exercise 4.1.3. Show that if {Ii; h~i~N. t~;9; is an independent set ofrandom variables, if fIJi is a Borel measurable function in Rile;, 1 ~ i ~ N, then

is independent. [Hint: Use the result in Exercise 4.1.1iii. 211.] Example 4.1.2. Let I and g be the functions for which the graphs are given in Figure 4.1.1 (f = g on (-oo,d]).

Section 4.1. Independence

213

y·axis 0
112

112

III

I-d

b-a

b-c

d-c

--=1=-=---=--

k2

------------

y = f(x)

o

a

b

c

d

1

x·axis

Figure 4.1.1.

Then Ind ({f}) = Ind ({g}) = Const. The proof is based on the metric density theorem [Rud): THEOREM. IF E E C THEN THERE IS IN E A NULL SET N SUCH THAT IF x E E\N THEN lim A (E n (x - e, x £!O 2e

+ e))

= 1.

Assume B is a Borel set in [0,1] and that {B,J ~f 1- 1 ([a,,8])} is independent. If < a < ,8 < kl' then J is the union of three equally long and disjoint intervals Ii resp. 12 resp. 13 that are subintervals of [a, b] resp. [b, c] resp. [c, d] and P( J) = 3P (lj), 1::; i ::; 3. Thus

°

P(B)P(J) = P(B n J) = P (B nIl)

°

+ P (B n 12) + P (B n 13) .

If 1,8 - al --+ 0, < a < ,8 < kt, then metric density arguments show that the only possible values for P(B) are 0, 1,~, 1. On the other hand, if kl < a < ,8 < k2 and 1,8 - al --+ then similar arguments show that the only possible values for P(B) are O,!, 1. Hence, modulo null sets, B must be [0, 1] or 0. Hence Ind(f) = Const. Arguments of the same kind show that Ind(g) = Const. [Note 4.1.1: If 1 and g are functions for which the conclusions above are valid, validity persists if 1 and g are replaced by homothetic translates of infinitely differentiable functions and g,

°

i

Chapter 4. Probability Theory

214

constructed by smoothly rounding the corners of the graphs of I and g. Since kl and k2 are arbitrary it follows that if p is a polynomial over JR then an arbitrarily small part of the graph of p can be excised and replaced by the graph of a translate of a smoothed homothetic version g of 9 so that the result p is a function again in Coo ([0,1], JR) and Ind (p) is again Const. Hence in each of the familiar function spaces built on [0,1], e.g., C ([0,1], JR), V ([0, 1], JR) et aI., there is a dense subset D consisting entirely of functions I such that Ind(f) = Const. Thus if S ~f {fn}neN is an independent sequence of random variables and In -+ I in any of a number of different modes of convergence, e.g., uniform, in V, pointwise, etc., then each In may be replaced by a gn such that gn -+ I in the same mode and yet for each n, Ind (gn) = Const. In other words, although the independence of a sequence of random variables is frequently an important part of the basis for a conclusion about the convergence of the sequence and the nature of the limit I, e.g., in the central limit theorem, I may well be the limit (in the same mode of convergence) of a sequence that is as far from independence as it is possible to be.] The set of trigonometric functions spans a dense subset of each of the familiar function spaces. Yet, as the next result shows, neither this set nor any other orthogonal set that spans a dense subset of one of these spaces can be independent. THEOREM 4.1.1. LET (X, S, P) BE A PROBABILISTIC MEASURE SITUATION SUCH THAT L2(X, JR) IS A Hilbert SPACE 'H. OF DIMENSION NOT LESS THAN 3. IF S ~f {f~heA IS INDEPENDENT, ITS CLOSED SPAN M IS SUCH THAT THE orthogonal complement Ml. IS NOT {O}. IF 1t IS INFINITE-DIMENSIONAL THEN Ml. IS ALSO INFINITE-DIMENSIONAL. PROOF. Since each

b is integrable, mathematical induction shows

that

{g~ ~f b -

Ix

b(x) dPheA

Ix

is independent and g~(x) dP = 0, A E A. Thus it may be assumed that for all A, b dP = 0. Finally it may be assumed that no I~ is constant

Ix

and the question is whether E ~f S U {I} can span 'H.. From (4.1.1) page 211 it follows that each product ¢ ~f bl ... I~n of pairwise different members of S is a nonzero element of 1t and is orthogonal to any b different from each of the factors. On the other hand, if I is one

Section 4.1. Independence

215

of the factors, say 1>•• , of ¢ and if n ~ 2 then (4.1.1) page 211 implies (1)•• , ¢) = O. In other words, M denoting the closed linear span of E, if n ~ 2 then ¢ E M 1. •

D If X consists of two elements

[ Remark 4.1.1:

Xl,

X2 and if

II == 1, h(Xl) = -h(X2) = 1, P({xd) = P({X2}) = 0.5 then {II, h} is an independent set consisting of orthonormal functions that span 'H. Hence the number 3 in the statement of THEOREM 4.1.1 is optimal.) On [0, 1) the Rademacher functions are defined as follows: def {

rn(X) =

1 sgn(sin211'2nx)

if n = 0 if n EN'

Exercise 4.1.4. Show that the set of Rademacher functions (random variables) is a maximal independent set for the measure situation ([0,1), C, "\), i.e., that if a nonconstant random variable f is adjoined to the set of Rademacher functions the resulting set is not independent. [Hint: To prove the maximality of the set it may be assumed that A ~f f- l ({I}) =F 0. Then if n > 1,

..\ (r;l ({I}) n A)

=

~"\(A).

The form of r;l ({I}) and the metric density theorem imply that "\(A) is unbounded.)

The construction in the proof of THEOREM 4.1.1 is related to the construction of the Walsh functions from the Rademacher functions [Zy]. Since the Walsh functions constitute a complete orthonormal set in £2 ([0, 1), C) there arises the question: Does the general construction above always yield a complete orthonormal set, at least if the set E is a maximal independent set? Example 4.1.3. Let f be a random variable f such that Ind(f) = Const. It follows that El ~f {J} UConst is a maximal independent set. The general construction used for E and applied to El leads to a an orthonormal set containing no more than three elements and thus, if the associated Hilbert space is of dimension at least four, the orthonormal set that emerges is not complete. In [Ge6] there is a more extensive discussion of independence phenomena.

Chapter 4. Probability Theory

216 4.2. Stochastic Processes

For a probabilistic measure situation (X, S, P) and a set {f~ hEA of random variables in IRx there is the set

of associated distribution /unctions. Each F~l, ... ,~n is in IRRn and

It follows that the functions in];" satisfy the five Kolmogorov criteria [Ko2]: i. if 1 ~ i ~ n then limzil-oo F~l,. .. ,~n (Xl, ... ,Xn ) ii. limz1loo, ... ,zn 100 F~l'''''~n (Xt, .. . , Xn) = Ij iii. for each i,

= OJ

Xi ~ X~ ~ F~l' ... '~n (Xl, ... , Xi,· .. ,Xn) ~ F~1! ... '~n (Xl, ... , X~, . .. ,Xn) j

iv. if {it, i 2 , ••• ,in} is a permutation of {I, 2, ... ,n} then

v. if k < n then

Kolmogorov showed that conversely, if a set];" satisfying his criteria is given then there is definable on a O'-algebra Z in IRA a probability measure P, i.e., a measure situation (IRA, Z, and random variables Ip. : IRA 3 (X~hEA 1-+ xp. for which];" constitutes the set of distribution functions. The O'-algebra Z is generated by the set of all cylinder sets based on Borel sets in IRn , n E N: a typical cylinder set has the following form:

p),

Z~1!""~n (A) ~f {(X~hEA : (X~hEA E IRA, (X~p ... , X~n) E A }

,

A a Borel set in IRn.

When A is a comer of the form {(at, ... , an) : ai ~ Xi, 1 ~ i probability P [Z~l, ... '~n (A)] of the corresponding cylinder set is

~

n} the

Section 4.2. Stochastic Processes

217

Thus the value at (Xl. . .. ,xn ) of the joint distribution function of the random variables h.p 1 ~ i ~ n, is FA1, ... ,An (Xb"" xn) as required. The extension of P to Z follows readily [Ko2]. It should be noted that 5 ~f IRA is a vector space of functions. The evaluation map: ¢ ~ ¢ (Ai) E IR taking the function ¢ in 5 to its value at Ai is a special kind of linear functional on S. The natural extension of this observation leads to the next construction when a) A is itself an ingredient of a measure situation (A, T, J.t) or b) when A is a topological space or c) when A is an n-dimensional manifold. For a) there are considerations of spaces £P (A, IR), 1 ~ P ~ 00; for b) there are considerations of C (A, IR); for c) there are considerations of vector spaces of functions satisfying differentiability requirements. In each instance there is the question of whether P induces a probability measure on the subspace to be studied. Stripped to essentials, the context is the following. i. Let V be a topological vector space, e.g., a Banach space, let (n, S, ll) be a probability measure situation, and let T be a linear map- of the

dual space V* of V into the set 'R of random variables on n. For each finite set {xi, ... ,x~} in V* and each Borel set A in IRn let Zxi ,... ,x;;;A

~f {x : x

E

V, (xi (x) , ... ,x~ (x))

E A}

be the cylinder set based on {xi, ... , x~; A}. ii. The set Z of all cylinder sets is an algebra on which one can define (modulo appropriate equivalences) the finitely additive set function J.t: Z 3 Zxi, ... ,x;;;A ~ II ({ w : wEn, (T(xi)(w), ... , T(x~)(w)) E A})

The description above is imprecise, e.g., J.t as defined depends on the n-tuple used to define Zxi, ... ,x;;;A and not on the cylinder itself. When suitable equivalence relations and their corresponding classes are brought into consideration the vagueness disappears. The interest here is in the following question: In what circumstances can J.t be extended to a count ably additive measure on Z, the a-algebra generated by Z? The answer is far from simple since it depends on V, on (n,S,ll), and on T. However, among the choices for A is the corner

in which case the measure of the corresponding cylinder set is the value Fxi, ... ,x;; (h, ...

,tn )

218

Chapter 4. Probability Theory

Since V may be viewed as a subset of lRV· the question above is reduced to whether P, restricted to V, yields a probability measure on the O'-algebra generated by the set of intersections of cylinder sets in lR v • with V. A not very helpful answer is the near-tautology: iff V is a thick subset of lR v·

[Halm]. In a more concrete fashion, Hemasinha [Hem] showed that if D is a bounded region in C, if is Lebesgue measure normalized on D so that n(D) = 1, if V is the Hilbert space of functions 1 harmonic in D and such that

n

L

I/(z)1 2 dO < 00,

and if T is any bounded endomorphism of V then the corresponding measure f..I. is countably additive. (For convenience and generality, Hemasinha worked with holomorphic functions defined on D.) On the other hand, if T is required to operate on a Hilbert space 'H. ~f which is its own dual, and, additionally, the endomorphism T is to map orthogonal pairs of functions into pairs of independent functions, the set function f..I. is not count ably additive. The following sequence of results yields the conclusion above. All functions considered are assumed to be in Li (0,0).

Li (0, n) of lR-valued functions,

LEMMA 4.2.1. IF 4J,'Y IS A PAIR OF ORTHONORMAL lR-VALUED FUNCTIONS THEN 4J ± 'Y IS A PAIR OF ORTHOGONAL FUNCTIONS. PROOF.

o LEMMA 4.2.2. IF BOTH {I, g} AND {I ± g} ARE SETS OF INDEPENDENT FUNCTIONS AND AT LEAST ONE OF THE FUNCTIONS IS NOT A CONSTANT, THEN EACH OF I,g, 1 + 9 AND 1 - 9 IS normally distributed, i.e., THE FORM OF THE DISTRIBUTION FUNCTION FOR EACH IS

-1.;2i0'

l

z

2

t exp(--2)dt

-00

20'

[Ge8]. PROOF. It may be assumed that

In = In In I(W)2 = In g(W)2 I(w)dn

dn

g(w)dn

=0

dO

= 1.

Section 4.2. Stochastic Processes

219

Let (f ± g) / /2 be h±. Because f and 9 are linear combinations of h± the result in Exercise 4.1.3. 212 implies that if

In exp(itf(w)) dll ')'(t) ~f In exp(itg(w)) dO 4J(t)

then

~f

{(t)

~f In exp(ith+(w)) dll = 4J (~) ')' (~)

l1(t)

~f In exp (ith_(w)) dll = 4J (~) ')' (~)

4J(t)

= { (~) 11 (~)

')'(t)

= {( ~) 11 (~).

Mathematical induction applied to the process of substituting the right members of the first two equations for the appearances of {(t) and l1(t) in the last two equations leads to the following formula:

4J(t)

=

nEN,

and a similar formula for ')'(t). Since, as t -. 0,

4J(t) =

')'(t)

=

(1 - t; + (t (1- t; +O(t2)) 0

2 ))

it follows that

4J(t)

= ')'(t) = exp ( _ t;) .

The inversion formula for Fourier transforms shows that f and 9 are normally distributed. Since f and 9 are independent, the distribution functions for f ± 9 are convolutions of the distribution functions for f and ±g. Since convolutions of normal distribution functions are themselves normal, the result follows.

D

Chapter 4. Probability Theory

220

Let T be an endomorphism of 'H. ~f Li(n, IT). Then T is said to induce a Gauf3ian measure JJ on the algebra Z of cylinder sets in 'H. if the joint distribution function for the set {T( Xl), ... , T( xn)} of independent random variables is of the form (4.2.1)

whence {T (Xl) , ... ,T (xn)} is independent. LEMMA 4.2.3. IF THE ENDOMORPHISM T OF THE INFINITE-DIMENSIONAL HILBERT SPACE 'H. ~ Li(n, IT) MAPS ORTHOGONAL FUNCTIONS INTO INDEPENDENT FUNCTIONS THEN T INDUCES A GAUSSIAN MEASURE ON THE ALGEBRA Z OF CYLINDER SETS IN 'H.. THIS GAUSSIAN MEASURE CANNOT BE EXTENDED TO A COUNTABLY ADDITIVE MEASURE ON THE O'-ALGEBRA Z GENERATED BY Z. PROOF. Since T maps orthogonal functions into independent functions it follows from LEMMA 4.2.2.218 that if {Xn}nEN is complete orthonormal set in 'H. then the finite-dimensional distribution functions for the random variables {T(Xn)}nEN take the form given in (4.2.1). If the Gauf.Uan measure JJ may be extended from Z to a countably additive measure on Z there emerge the following contradictory relations

[Kur]: i. if N 3 kn

l

00

then

00

'H. =

U{x n=l

00

X

UHnj

E 'H., I(x, Xi)1 $ n, 1 $ i $ kn } ~

n=l

ii. for any M in (0,1) the numbers kn can be chosen so that

iii. JJ ('H.) $ E:=l Mn = I~M' whence JJ ('H.) = OJ iv. 'H. = {x : (x, xI) E 1R} and so JJ ('H.) = (211')-! fIR exp ( -~) ds = 1.

o The reconciliation between the result above and Hemasinha's work stems from the fact that the functions in his model of Hilbert space are holomorphic, and, trivialities aside, sets of holomorphic functions cannot be independent [Ge6], i.e., in Hemasinha's model, there is no endomorphism T satisfying the hypotheses of LEMMA 4.2.3.

221

Section 4.3. Thansition Matrices

Example 4.2.1. In an infinite-dimensional Hilbert space 'Ii there can be no nontrivial Borel measure that is translation-invariant or unitarily invariant. Indeed if IL is a nontrivial Borel measure, let {tPn}nEN be an orthonormal set and let Bn be a ball centered at ~ tPn and of radius 0.1. If IL is translation-invariant or unitarily invariant, IL (Bn) > O. Since i :f: j => Bi n B j = 0 it follows that if B ~f {x : x E 'Ii, IIxll :5 1 } then B:::>

U

Bn and IL

nEN

(U

Bn)

= 00.

nEN

Hence the unit ball centered at 0 has infinite measure and, by a similar argument, every ball of positive radius and centered anywhere, has infinite measure and so, for every Borel set A, IL(A} = 0 or IL(A} = 00, i.e., IL is trivial. 4.3. Transition Matrices

A transition matrix P ~f

(Pij }~j~l

is characterized by the conditions

n

LPij

= 1,

1:5 i :5 n

j=l

Pij ~

0, 1:5 i, j :5 n.

The number Pij is interpreted as the probability that a system in "state" i will change into "state" j. For many transition matrices P it can be shown that · pn ~f P.00 11m n ..... oo

exists. For example, if for some k in N, all entries in pk are positive, then Poo exists [Ge9]. The matrix

is a transition matrix whereas

~) ~)

if k is odd if k is even,

whence Aoo does not exist. A clue to this behavior is found in an examination of the eigenvalues, ±1 of A. The Jordan normal form of A is

Chapter 4. Probability Theory

222

which immediately reveals why Aoo does not exist. For any transition matrix P, the vector (1,1, ... , l)t is an eigenvector corresponding to the eigenvalue 1, and for every eigenvalue ~, I~I $ 1. THEOREM 4.3.1. IF P ~f (Pij):'~;'!.l IS A TRANSITION MATRIX AND IF

* * = 1 OR * = 0,

1$ m $ M

ARE THE Jordan blocks OF P THEN Poo EXISTS IFF:

i. I~ml < 1 WHENEVER * = 1; ii. ~m = 1 WHENEVER I~m I = 1.

PROOF. If I~ml

= 1 then limk ..... oo ~~ exists iff ~m = 1, cf.

[Ge9].

D Exercise 4.3.1. Regard each n x n transition matrix P as a vector in 3 an . Show that the set l' of n x n transition matrices is the intersection of the nonnegative orthant a(n 3.+) and n hyperplanes. Exercise 4.3.2. View l' as a "flat" part of n 2 - n-dimensional Euclidean space and thus as endowed with the inherited Euclidean topology and Lebesgue measure ~n2-n ~f J.I. • Let 1'00 be the subset consisting of transition matrices P for which P00 exists. Show that

and that 1'\ 1'00 is a dense open subset of 1', cf. COROLLARY 1.3.1.1. 26.

5.

Foundations

5.1. Logic

From early times human language has been a source of counterexamples to the belief that normal discourse is consistent. The sentence, "This statement is false," can be neither true nor false. The phrase "not self-descriptive" is neither self-descriptive nor not self-descriptive. Can an omnipotent being overpower itself? In [BarE] there is an extensive discussion of those aspects of language that deal with grammatically accurate but logically daunting statements. Mathematical versions of such paradoxes, antinomies, explicitly or implicitly self-referential words and sentences, etc., eventually led to the search for a formal system of logic in which the perils of inconsistency are absent or at least so remote that humankind need have no fear of their obtrusion into scientific discourse. The next few paragraphs, summarizing the presentation in [Me], deal with the fundamental concepts of a formal system of logic F.

[ Note 5.1.1: However rigorous, however formal, however restrictive the formal systems themselves, the proving of theorems about these same systems inescapably leads to reliance upon the use of human language whence the problems first emerged. Thus, it appears, that in the drive to achieve consistency and to avoid paradox, the logicians resort to harshly restricted modes of reasoning that are no more formal than the modes that lead to the 223

224

Chapter 5. Foundations paradoxes, the antinomies, the self-referential sentences, etc. The hope that success will crown the effort rests on the "finitism" of the approach. The next paragraph, introducing the formalization of logic, adverts almost immediately to a "countable set" without defining a countable set. Presumably a countable set is a set (not defined) that can be put in bijective correspondence (not defined) with N (also not defiried). Later developments of formal logic and set theory lead to an axiomatic formalization of N and its consequent structures, Z, Q, JR, C, lBl, et a1. Is there no circularity in the procedure? For a profound discussion of these matters the reader is urged to consult [HiB].]

There is a countable set S of symbols, finite sequences of which are expressions. Some of the symbols are logical connectives such as V ("or"), 1\ ("and"), -+ ("implies"), and..., ("negation"). Others are quantifiers V ("for all"), 3 ("there exists"), function letters f, g, ... , predicate letters P, Q, . .. , variables x, y, ... , and constants a, b, . ... A predicate P or a function f always appears in association with a nonempty set consisting of finitely many predicates, constants and variables ("arguments"), e.g., P(a), f(x,y,P). A quantifier always appears in association with variables and predicates, e.g., V(x)P(P,Q,x,y,a,b,c). A large part of formal logic, in particular the part discussed below, is devoted to the study of first-order theories in which the arguments of predicates may not be predicates or functions and in which the argument of a quantifier must be a variable. Thus in first-order theories forms such as 3(P)(P -+ Q) are not included. Within the set of expressions there is a subset WF consisting of wellformed formulae (wfs) and a subsubset A consisting of those wfs that are the axioms. There is a finite set R of rules of inference that permit the chaining together of axioms to lead to consequences and the chaining together of axioms and/or consequences to produce proofs. The last link in a proofchain is a theorem, (which might be an axiom). The objects above constitute a framework in terms of which specific mathematical entities, e.g., groups, N, etc., can be discussed by adding to the logical symbols and axioms other symbols and axioms. For groups the symbols and axioms in Subsection 1.1.1 are the added objects. For formal number theory, i.e., for the treatment of Z, the symbols and axioms added are some carefully tailored version of those given originally by Dedekind but known more popularly as Peano's axioms. Closely associated with a formal system :F are interpretations and models for it. An interpretation is a "concrete" nonempty set D and assignments: i. of each n-variable predicate to a relation in D, i.e., to a subset of Dnj

Section 5.1. Logic

225

zz. of each n-variable function to a function D n iii. of each constant to a fixed element of D.

1-+

Dj

The symbols ..." -, 'V, and 3 are given their "usual" meanings. There are systematic definitions (due to Tarski) of the notions of satisfiability and truth of wfs. Informally, a wf A is satisfiable lor some interpretation I, if A obtains for some substitution in A. For example, in group theory, if the interpretation 1 is the set of nonzero real numbers regarded as an abelian group with respect to multiplication, then the wf A ~f {x 2 = I} is satisfiable iff one substitutes for x the number 1 or the number -1. On the other hand, the same wf A (written additively {2x = I}) is not satisfiable in Z regarded as the abelian additive group of integers. A wf is satisfiable iff it is satisfiable in some interpretation. Again informally, a wf A is true in an interpretation 1 if A obtains for every substitution. For example the wf A ~f x + x = e obtains in Z2 for all (both) substitutions x 1-+ 0 and x 1-+ 1. A wf A is logically valid iff A is true in every interpretation. There are natural (informal) definitions of contradictory wfs, of the phrase A implies B, and of the phrase A is equivalent to B. An interpretation 1 is a model M(I) for a set of wfs iff each wf is true for I. In the language and context of the outline above, Godel, who was soon to become the pre-eminent logician among his contemporaries, proved the formal equivalence of the notions of theorem and logical validity.

A

GODEL'S COMPLETENESS THEOREM. IN A FORMAL SYSTEM IS A THEOREM IFF A IS LOGICALLY VALID [Gol].

:F A

WF

In [Gol] Godel proved a more striking result: GODEL'S COUNTABILITY THEOREM.

EVERY CONSISTENT FIRST-OR-

DER SYSTEM HAS A COUNTABLE MODEL.

A consequence of Godel's count ability theorem is a result proved earlier by Lowenheim [Low] and Skolem [Sk]. LOWENHEIM-SKOLEM THEOREM. IF A FIRST-ORDER THEORY HAS A MODEL IT HAS A COUNTABLE MODEL.

[ Note 5.1.2: Do the Godel-Lowenheim-Skolem results imply that, despite what every mathematician knows, IR is countable? A simple answer is "No!" The reason lies in the subtlety of the notion of model. In the countable model of the formal system for analysis the "uncountability" of IR is the assertion that for the D of the interpretation there is no map I : D 1-+ D such that in the model I (N) = JR.] The mechanism above having been established, its founders planned to produce a formal system :F adequate to deal at least with number theory, i.e., to cope with theorems about Z. In this system each wf A or its negation

226

Chapter 5. Foundations

...,A was to be a theorem and not both A and ...,A were to be theorems (the latter desideratum was for consistency). Godel and Rosser [Go2, Ross] proved that any consistent formal logical system :F that deals with N contains undecidable wfs. No formal proof exists for each nor for its negation: :F is incomplete. One among those undecidable wfs, has a striking self-referential interpretation: "The system :F is consistent." [ Note 5.1.3: Since the wf A interpreted above is undecidable it may be adjoined to :F to form a new system :F which is as consistent as:F. But then there is in :F an undecidable wf A' interpretable, like A, as asserting that :F is consistent. In:F the wf A is an axiom, hence is a theorem, and thus is decidable.] One view of Godel's incompleteness theorem is the following. If one can consistently axiomatize logic so that there are mechanical rules whereby one passes, step-by-step, from axioms to theorems then one can imagine a machine that systematically lists all proofs, e.g., proofs involving one step, proofs involving two steps, etc. In theory the machine creates a count ably infinite list of all theorems, each preceded by its proof. Then if a wf Tis given, the list can be consulted to determine whether 7 or ...,7 appears in the list of theorems. To determine whether Tor ...,Tis in the list, the machine is programmed in some way, e.g., to compare 7 and then ...,7 with each listed theorem. The original hope of the axioma.tizers was that there is a program that, given a wf T, checks 7 and then ...,7 against each of the listed theorems and, in finitely many steps, finds either Tor ...,To There arises the question of whether the machine, however programmed to carry out the task, will, for a given wf, ever stop. Godel's result says in effect that if the axiomatized system :F is consistent and deals with theorems about N then there is a wf for which the machine will never stop. Neither the wf nor its negation will appear on the list of derivable theorems. There is a wf 8 and its negation ...,8. For any N in N, the machine, having compared both 8 and ...,8 with each of the first N theorems in the list, will have encountered neither 8 nor ...,8. Hence at no stage of the process will there be a decision that 8 is a theorem or that ...,8 is a theorem: 8 is an undecidable formula: :F is incomplete. There are various ways for coding or numbering wfs, proofs, theorems, etc. There are various ways for coding or numbering programs for machines. Each such coding method assigns to each wf, proof, theorem, or program a natural number. Such a coding can be prepared so that each natural number is the code for some wf and each natural number is also the code for some program. Godel's conclusion, says that there is a wf, say numbered n, such that for any checking program, say numbered m, the machine, using program m to check wf n (and the negation of wf n) against the list of theorems, will

Section 5.1. Logic

227

never halt. The flavor of his argument can be conveyed in the following way by considering an analogous problem in computer operation. Every computer program is ultimately a finite sequence of zeros and ones. Similarly, every data-set is also a finite sequence of zeros and ones. Since there are count ably infinitely many programs and countably infinitely many data-sets, the programs may be numbered 1,2, ... , and the data-sets may be numbered 1,2, .... Some programs applied to some data-sets stop after performing finitely many steps, others never stop. For example, the simplex method applied to some PLPP cycles endlessly. It is conceivable that, for a given pair (m, n) representing a program numbered m and a data-set numbered n, one can determine, say via some TESTPROGRAM whether program m, applied to data-set n, halts or fails to halt. In other words: Confronted with any pair (m, n), TESTPROGRAM processes the pair and reports EITHER that program m applied to data-set n stops after finitely many steps OR that program m applied to data-set n never stops. The next discussion shows that no such TESTPROGRAM exists. If TESTPROGRAM exists one may assume that TESTPROGRAM calculates the value of a function! : N x N 3 (m, n) 1-+ !(m, n) such that:

= 0 if program numbered m applied to data-set numbered n stops; ii. !(m, n) = 1 if program numbered m applied to data-set numbered n never stops. i. !(m, n)

In the list of all programs there is one, STOPGO, numbered, say ms, and operating as follows.

iii. Given the number n, first STOPGO calculates !(ms, n). iv. If !(ms, n) = 1, then STOPGO prints the number 2 and stops. v. If !(ms, n) = 0 STOPGO engages in the task of printing the sequence of markers in the binary representation of 1f'. Thus if TESTPROGRAM reports that STOPGO (program ms) applied to data-set n never stops, i.e., if !(ms, n) = 1, then STOP GO applied to data-set n stops. If TESTPROGRAM reports that STOPGO (program ms again) applied to data-set n stops, i.e., if !(ms, n) = 0, then STOPGO never stops. It follows that there is no program like TESTPROGRAM that can accurately decide about all pairs (m, n) whether program m applied to data-set n stops. The conclusion reached is interpreted as follows: There is no algorithmic, systematic technique, defined a priori, that can be used to determine for each wf T whether T or ~T is derivable as a theorem.

Chapter 5. Foundations

228

The technicalities of rigorously formalizing the discussion above are lengthy but straightforward. Excellent sources for the details are [Me, Rog]. There is an illuminating discussion of these matters in [Jo]. Godel's work gave rise to the study of recursion, the definition of such terms as algorithm, effectively computable, Turing machine, ... , and a host of related topics and concepts. Theorems of varying degrees of strength and impressiveness emerged. It is the opinion of many that the result about the nonexistence of a TESTPROGRAM typifies the field. It is viewed as the unsolvability of the halting problem. The work of Church, Godel, Herbrand, Kleene, Post, and Turing all drove to the same conclusion that consistent formal logical systems rich enough to deal with N are perforce incomplete in that they contain meaningful and yet undecidable wfs. Following their work many others showed the undecidability of many "natural" wfs in mathematics, e.g., the wf corresponding to the word problem in finitely presented groups (cf. Note 1.1.5.2. 11). In [Bar, BarE, Chait, Chai2, Davit, Davi2, Kin, Lam, Me, Ross, T, TaMR, Tor] there is more information on the topics discussed above. It should be noted that once undecidability surfaced, all sorts of questions were attacked. An example is Hilbert's tenth problem. A Z-polynomial P is, for some n in N and a set .}.11, ... ,ln_ . <M { a'11,···,1"

consisting of n-tuples of integers, the map

P : Zn 3 (Xl, ... ,Xn )

t-+

L

ailloo.,inxll ... x~n E Z.

i1, ... ,i n

Let 1) be the set of all Z-polynomials. Is there an algorithm such that for each P in 1) the algorithm determines in finitely many steps (the number of steps depending on the polynomial P) whether the Diophantine equation

· so1u t IOn s def = ( 81, ..• , 8 n ).In fU'1In?• Matijasevic [Mati, Mat 2] in 1970 and 1971 showed that no such algorithm exists. Of somewhat independent interest was a fortuitous discovery by von Neumann. In the course of writing [N3] on operator theory, he could have used a rather general proposition about the measurability of images of analytic sets [Kur]. He showed that the particular image of the analytic set

has

a

Section 5.2. Set Theory

229

under consideration was measurable but he noted that the general proposition regarding the measurability of all such sets is undecidable. One may speculate that, e.g., Fermat's theorem T is undecidable in the axiomatic framework for N. If that is the case, then for all practical purposes, Tis true, since any counterexample to the statement of Twould constitute a proof of the ...,T and thereby demonstrate that T is decidable. If Tis undecidable never will there be found nonzero integers x, y, z and a natural number n greater than 2 such that xn + yn = zn. [Note 5.1.4: As recently as 1989, a new and apparently shorter proof of Godel's undecidability theorem was offered by Boolos [Bo]. After examining that proof, Professor Richard Vesley at the State University of New York at Buffalo made the following observations [V]:

z. Call an algorithm correct if it never lists a false theorem. A truth omitted by an algorithm is a true wf not listed by the algorithm. n. Boolos's argument shows that if M is a correct algorithm then there is a truth omitted by M. iii. Godel's original work produced an algorithm Ml that, applied to any correct algorithm M of a restricted class of algorithms, yields a truth omitted by M. Subsequently there was produced an algorithm M2 that, applied to any correct algorithm M yields a truth omitted by M. The thrust of Vesley's comments is that Boolos's proof is existential and nonconstructive. On the other hand, Godel's proof is constructive in the sense that it describes the algorithm Ml that can be applied to any correct algorithm M and thereby exhibit a truth omitted by M. (What would Ml applied to Ml yield?)]

5.2. Set Theory

Closely related to the problem of formalizing logic is the problem of axiomatizing set theory. Current thinking has settled On the Zermelo-Fraenkel (ZF) formulation of the basic axioms for a theory of sets [Me]. These axioms, related to a more general system NEG proposed by VOn Neumann and modified in stages by Bernays and Godel, involve objects called classes and only one predicate, symbolized by E, intended to suggest "membership." Among the classes there are sets distinguished as follows: a class X is a set iff there is a class Y such that X E Y. Customarily, sets are denoted by lower case letters, classes by capital letters. Every set is a class but not every class is necessarily a set. In terms of E there are defined relations C (inclusion), ~ (proper inclusion), and = (equality) among sets. The axioms provide for an empty set

0, for subsets

Chapter 5. Foundations

230

y of set x, for the power class 2x of any class X, for the Cartesian product X x Y of two classes X and Y, etc. Ordinal numbers are defined without reference to the Axiom of Choice or any of its logical equivalents, the Axiom of Zermelo, i.e., the Well-ordering Principle, etc. [Note 5.2.1: The G8del-L8wenheim-Skolem theorem as it bears on the uncountability of R may be viewed as follows. To say that R is uncountable is to say that there is no surjection 1 : N t-+ R. A surjection, like any map, may be regarded as the graph of the map in D x D. To say that 1 does not exist is to say that the graph of 1 does not exist, i.e., that there is in D x D no set, as distinguished from a class, that can serve as the graph of the surjection in question. In the countable model for analysis, the countable set R is not the surjective image of N, hence in the countable model for analysis, the countable set representing R is not countable in the language of the model.] Once these axioms are accepted, extensions are considered, so that, e.g., the Axiom 01 Choice C, the Continuum Hypothesis CH, the Generalized Continuum Hypothesis GCH may be added to the axioms of ZF. The corresponding axiom systems are ZFC, ZFCCH, ZFCGCH, etc. In 1940 G8del [Go3] showed that if ZF is consistent then the three extensions cited are also consistent. Finally, in 1963 Cohen [Cohl, Coh2, Coh3] showed that C, CH, and GCH are independent of ZF, in other words, C, CH, and GCH are undecidable propositions in ZF, i.e., ZF with anyone or more of C, CH, and GCH adjoined is just as consistent as ZF with anyone or more of -.C, -.CH, and -.GCH adjoined. Cohen invented a new technique, lorcing, whereby, starting with a consistent model of ZF, he replaced the model (via forcing) by larger consistent models in which various consistent combinations of C, -.C, CH, -.CH, GCH, and -.GCH obtain. Excellent references for this topic are [Bar, Coh3, Je]. Following upon Cohen's accomplishment, Solovay [Sol] showed that ZF may be extended in another consistent way by adding the following axiom: AXIOM OF SOLOVAY. EVERY FUNCTION 1 : Rn t-+ R IS LEBESGUE MEASURABLE.

Since the Axiom of Choice implies the negation of the Axiom of Solovay it follows that, although both ZFC and ZFS are consistent, they are mutually incompatible axiom systems. If ZFS replaces ZFC as a basis for set theory there arises the following situation. Let the topology of a topological vector space V be defined by a separating and filtering set P ~f {P~hEA of seminorms, i.e., V is a separated locally convex vector space, LCV. Define such a topological vector space V

Section 5.2. Set Theory

231

as good if every seminorm 7r defined on V is continuous in the sense that there is a constant C ff such that for each p)"

x E V:::::} 7r{x)

~ CffPA{X).

Most of the familiar locally convex (topological) vector spaces are good. Garnir showed that if ZFS is used instead of ZFC then every linear map T : V 1--+ W of a good space V into a locally convex vector space W is continuous [Gar].

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SUPPLEMENTAL BIBLIOGRAPHY

The list below Wag compiled by Professor R. B. Burckel and it is with his kind permission that the items are included in this volume. The authors are grateful for his generosity and scholarship. Bauer, W.R. and Benner, R.H. - The nonexistence of a Banach space of countably infinite Hamel dimension, Amer. Math. Monthly, 78, (1971), 895--6. Benedicks, M. - On the Fourier transforms of functions supported on sets of finite Lebesgue measure, Jour. Math., Anal. and Applications, 106, (1985), 180-3. Broadman, E. - Universal covering series, Amer. Math. Monthly 79, (1972), 780-l. Brown, A. - An elementary example of a continuous singular function, Amer. Math. Monthly, 76, (1969), 295-7. Bruckner, A. - Some new simple proofs of old difficult theorems, Real Analysis Exchange, 9, (1984),63-78. Cantor, R., Eisenberg, M., and Mandelbaum, E.M. - A theorem on Riemann integration, Jour. Lon. Math. Soc., 37, (1962), 285--6. Cater, F.S. - Most monotone functions are not singular, Amer. Math. Monthly, 89, (1982), 466-9. - - - - Functions with prescribed local maximum points, Rocky Mountain Jour. of Math., 15, (1984), 215-7. Differentiable nowhere analytic functions, Amer. Math. Monthly, 91, (1984), 618-24. - - - - Equal integrals of functions, Can. Math. Bull., 28, (1985), 200-4. - - - - Mappings into sets of measure zero, Rocky Mountain Jour. of Math., 16, (1986), 163-7l. - - - - An elementary proof of a theorem on unilateral derivatives, Canadian Math. Bull., 29, (1986), 341-3. Conway, J. - The inadequacy of sequences, Amer. Math. Monthly, 76, (1969), 68-9. Darst, R. and Goffman, C. - A Borel set which contains no rectangles, Amer. Math. Monthly, 77, (1970),728-9. De Guzman, M. - Some paradoxical sets with applications in the geometric theory of real variables, L'Enseignement de Math., (2), 29, (1983), 1-14. Donoghue, W. F. Jr. - On the lifting property, Proc. Amer. Math. Soc., 16, (1965), 913-4. Dressler, R.E. and Kirk, R.B. - Non-measurable sets of reals whose measurable subsets are countable, Israel Jour. of Math., 11, (1972),265-70. 243

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Supplemental Bibliography

Drobot, V. and Morayne, M. - Continuous functions with a dense set of proper local maxima, Amer. Math. Monthly, 92, (1985), 209-11. Dubuc, S. - Courbes de von Koch et courbes d'Osgood, C.R. Math. Rep. Acad. Sci. Canada, 5, (1983), 173-8. Edwards, D.A. - On translates of LOO-functions, Jour. Lon. Math. Soc., 36, (1961), 431-2. Erdos, P. and Stone, M.H. - On the sum of two Borel sets, Proc. Amer. Math. Soc., 25, (1970),304-6. Fremlin, D.H. - Products of Radon measures: a counterexample, Canadian Math. Bull., 19, (1976), 285-9. Gaudry, G.!. - Sets of positive product measure in which every rectangle is null, Amer. Math. Monthly, 81, (1974),889-90. Gillis, J. - Some combinatorial properties of measurable sets, Quart. Jour. Math., 7, (1936), 191-8. - - - - Note on a property of measurable sets, Jour. Lon. Math. Soc., 11, (1936), 139-41. Goffman, C. - A bounded derivative which is not Riemann integrable, Amer. Math. Monthly, 84, (1977), 205-6. Goffman, C. and Pedrick, G. - A proof of the homeomorphism of Lebesgue-Stieltjes measure with Lebesgue measure, Proc. Amer. Math. Soc., 52, (1975), 196-8. Goldstein, A.S. - A dense set in L 1(-00,00), Amer. Math. Monthly, 85, (1978), 687-90. Hanisch, H., Hirsch, W.M., and Renyi, A. - Measure in denumerable spaces, Amer. Math. Monthly, 76, (1969),494-502. Hausdorff, F. - Uber halbstetige Funktionen und deren Verallgemeinerung, Math. Zeit., 5, (1919), 292-309. Henle, J .M. - Functions with arbitrarily small periods, Amer. Math. Monthly, 87, (1980), 816; 90, (1983), 475. Henle, J .M. and Wagon, S. - A translation-invariant measure, Amer. Math. Monthly, 90, (1983),62-3. Hong, Y. and Tong, J. - Decomposition of a function into measurable functions, Amer. Math. Monthly, 90, (1983), 573. Jamison, R.E. - A quick proof for a one-dimensional version of Liapounoff's theorem, Amer. Math. Monthly, 81, (1974), 507-8. Johnson, B.E. - Separate continuity and measurability, Proc. Amer. Math. Soc., 20, (1969), 420-2. Johnson, G.W. - An unsymmetric Fubini theorem, Amer. Math. Monthly, 91, (1984), 131-3. Katznelson, Y. and Stromberg, K. - Everywhere differentiable nowhere monotone functions, Amer. Math. Monthly, 81, (1974), 349-54.

Supplemental Bibliography

245

Kaufman, R. and Rickert, N. - An inequality concerning measures, Bull. Amer. Math. Soc., 72, (1966), 672-6. Kirk, R.B. - Sets which split families of measurable sets, Amer. Math. Monthly, 79, (1972), 884-6. Knopp, K. - Einheitliche Erzeugung und Darstellung der K unJen von Peano, Osgood und v. Koch, Archiv der Math. u. Physik 26, (1917), 10314. Leech, J. - Filling an open set with squares of specified areas, Amer. Math. Monthly, 87, (1980), 755-6. Leland, K.O. - Finite dimensional translation invariant spaces, Amer. Math. Monthly, 75, (1968), 757-8. Leonard, J.L. - On nonmeasurable sets, Amer. Math. Monthly, 76, (1969), 551-2. Lewin, J. W. - A truly elementary approach to the bounded convergence theorem, Amer. Math. Monthly, 93, (1986), 395-7; 94, (1987), 988-93. Machara, R. - On a connected dense proper subgroup of JR2 whose complement is connected, Proc. Amer. Math. Soc., 91, (1986), 556-8. Mattics, L.E. - Singular monotonic junctions, Amer. Math. Monthly, 84, (1977), 745-6. Milcetich, J. - Cartesian product measures, Amer. Math. Monthly, 78, (1971),550-1. Miller, A.D. and Vyborny, R. - Some remarks on junctions with onesided derivatives, Amer. Math. Monthly, 93" (1986),471-5. Miller, W.A. - Images of monotone junctions, Amer. Math. Monthly, 90, (1983), 408-9. Moran, W. - Separate continuity and supports of measures, Jour. Lon. Math. Soc.,44, (1969),320-4. Mussman, D. and Plachky, D. - Die Cantorsche Abbildung ist ein Borel-Isomorphismus, Elemente der Math., 35, (1980),42-3. Newman, D.J. - Translates are always dense on the half line, Proc. Amer. Math. Soc., 21, (1969), 511-2. Overdijk, D.A., Simons, F .H., and Thiemann, J .G.F. - A comment on unions of rings, Indagationes Math., 41, (1979),439-41. Pelling, M.J. - Borel subsets of a product space, Amer. Math. Monthly, 90, (1983), 136-8. Pettis, B.J. - Sequence with arbitrarily slow convergence, Amer. Math. Monthly, 68, (1961), 302. Randolph, J.F. - Distances between points of the Cantor set, Amer. Math. Monthly,47, (1940),549-51. Rao, B.V. - Remarks on vector sums of Borel sets, Colloq. Math., 25, (1972), 103-4 and 64.

246

Supplemental Bibliography

Rogers, C.A. - Compact Borelian sets, Jour. Lon. Math. Soc., 2, (1970), 369-71. Rosenthal, J. - N onmeasurable invariant sets, Amer. Math. Monthly, 82, (1975), 488-91. Rubel, L.A. and Siskakis, A. - A net of exponentials converging to a nonmeasurable /unction, Amer. Math. Monthly, 90, (1983),394-6. Rudin, W. - An arithmetic property of Riemann sums, Proc. Amer. Math. Soc., 15, (1964), 321-4. - - - - Well-distributed measurable sets, Amer. Math. Monthly, 90, (1983), 41-2. Russell, A.M. - Further comments on the variation /unction, Amer. Math. Monthly, 86, (1979), 480-2. Salat. T. - Functions that are monotone on no interval, Amer. Math. Monthly, 88, (1981), 754-5. Sinha, R. - On the inclusion relations between Lr (I') and £B (1'), Indian Jour. Pure and Appl. Math., 13, (1982), 1046-8. Stromberg, K. - An elementary proof of Steinhaus's theorem, Proc. Amer. Math. Soc., 36, (1972), 308. Takacs, L. - An increasing singular continuous /unction, Amer. Math. Monthly, 85, (1978), 35-7. Thomas, R. - A combinatorial construction of a nonmeasurable set, Amer. Math. Monthly, 92, (1985),421-2. Trautner, R. - A covering principle in real analysis, Quart. Jour. of Math., Oxford, 38, (1987), 127-30. Tsing, N .K. - Infinite-dimensional Banach spaces must have uncountable basis - an elementary proof, Amer. Math. Monthly, 91, (1984),505-6. Villani, A. - Another note on the inclusion V(JL) C Lq(JL), Amer. Math. Monthly, 92, (1985),485-7. Walker, P.L. - On Lebesgue integrable derivatives, Amer. Math. Monthly, 84, (1977), 287-8. Walter, W. - A counterexample in connection with Egorov's theorem, Amer. Math. Monthly, 84, (1977), 118-9. Wesler, O. - An infinite packing theorem for the sphere, Proc. Amer. Math. Soc., 11, (1960), 324-6. Weston, J.D. - A counterexample concerning Egoroff's theorem, Jour. Lon. Math. Soc., 34, (1959), 139-40; 35, (1960), 366. Wilker, J.B. - Space curves that point almost everywhere, Trans. Amer. Math. Soc., 250, (1979), 263-74. Young, R.M. - An elementary proof of a trigonometric identity, Amer. Math. Monthly, 86, (1979),296. Zaanen, A.C. - Continuity of measurable /unctions, Amer. Math. Monthly, 93, (1986), 128-30.

Supplemental Bibliography Zaji~ek,

L. -

247

An elementary proof of the one-dimensional density theo-

rem, Amer. Math. Monthly, 86, (1979), 297-8.

Zamftrescu, T. - Most monotone functions are singular, Amer. Math. Monthly, 88, (1981), 47-8. Zolezzi, T. - On weak convergence in L oo , Indiana Univ. Math. Jour., 23, (1974), 765-6.

SYMBOL LIST

The notation a.b.c. d indicates Chapter a, Section b, Subsection c, page dj similarly a.b. c indicates Chapter a, Section b, page c.

A 5.1. 224: the set of axioms of a formal logical system. AO 2.1.1. 45: the interior of the set A. AE 2.3.4. 170: the set of algebraic complex numbers in the set E. An 1.1.4. 8: the alternating group, i.e., the set of even permutations, on {I, ... ,n}. {A} 1.2.2.21: for an algebra A the associated algebra in which multiplication: A x A 3 (a, b) ........ ab is replaced by a new multiplication: A x A 3 (a,b) ........ aob~f ab-ba. (aij)W~1 1.3.1. 25 AA-1 1.1.4. 5: for a set A in a group G, the set

{ab- 1

:

a, bE A}.

See also X 0 Y. AC 2.1.2. 55, 2.1.2. 65: the set of Absolutely Continuous functions. a.e. 1.1.4. 7: almost everywhere. {A : P} 1.1.4. 6: the set of all A for which P obtains. A(S) 2.2.1. 118: the (surface) area of the set Sin JR3. B** 2.3.2. 164 B1 2.2.3. 144: the unit ball in JRn • BAP 2.3.1. 158: Bounded Approximation Property. BES 2.2.1. 118: the BESicovitch sphere. BV 2.1.2. 54: the set of functions of Bounded Variation. B(x) 2.2.2. 142 B(Y) 2.2.2. 142 B 2.2.1. 120: a homeomorphic image in JR3 of B1. the unit ball in JR3 j 2.3.1. 159: a (Schauder) basis for a Banach spacej 2.3.3. 168: the category of Banach spaces and continuous homomorphisms. B 2.3.2. 162: a biorthogonal set. Co 2.3.1. 159 c 2.3.2. 163 C 5.2.229 C 1.1.5. 18 (C,a) 2.1.3. 76 CON 2.3.1. 156: Complete OrthoNormal Set. 249

250

Symbol List

Const 4.1. 212: the set of constant functions. Cont(f) 2.1.1. 42 Conv 2.3.3. 168 Coo 2.1.1. 51, 2.1.2. 63 C (T, JR) 2.3.3. 165 C~k) (JR, C) 2.4.1. 172 C ([0,1], JR) 2.1.1.0 51 Co 1.1.4. 7: the Cantor set. Co (X, JR) 2.3.2. 163 C a 2.2.1. 115 Cp (JR, JR) 2.3.3. 165 Co 2.1.2. 55 3.2.2. 200: the discrete topology. V 1.3.1. 26: the set of diagonable matrices. ~f 1.1.2. 3: "(is) defined to be." deg(P) 2.1.3. 94: the degree of polynomial P. det 1.3.1. 26, 2.5.1. 177: determinant. diam(E): the diameter of the set E in a metric space (X, d). Diff(f) 2.1.1. 42 Discont(f) 2.1.1. 42, 2.1.1. 49, 2.1.2. 64 D(a,r) 2.6.1. 180 D(a, r)O 2.6.1. 180: {z : z E C, Iz - al < r}. V M 2.2.1. 112 F 2.3.1. 157: the Franklin system of orthogonal functions in C ([0, 1], C)j 5.1. 223: a formal system of logic. F 2.2.1. 105: the set of all closed subsets of a topological space. 2.4.2. 176: the n-fold convolution of the function 1 with itself. I(x) 2.2.2. 140 I(Y) 2.2.2. 140 FN 2.1.3. 74: Fejer's kernel. Fu 2.1.1. 43: the union of a countable set of closed sets. 9 1.1.4. 8: the category of groups. GCD 1.2.3. 23: Greatest Common Divisor. G/j 2.1.1. 43: the intersection of a countable set of open sets. G : H 1.1.2. 3: the index of the subgroup H in the group G. JH[ 1.1.5. 13: the set of quaternions. HEMIBES 2.2.1. 120: the BESicovitch HEMIsphere. H(O) 2.6.1. 180: the set of functions holomorphic in the region 0 (c C). Hom(A, B) 2.4.1. 172: the set of Homomorphisms: h: A ........ B. I 1.3.2. 36: [1il inv id 2.1.1. 52, 2.2.2. 143: the identity map. I 2.1.3. 68: the set of Irrational (complex) numbers. IE 2.3.4. 170: the set of Irrational numbers in the subset E of C. iff 1.1.2. 3: if and only if.

o

r·

Symbol List

251

im 1.1.3. 4 Ind( S) 4.1. 212: for a set S of random variables, the set of random variables f such that S U f is independent. K 1.1.4. 5, 2.2.1. 105: the set of compact subsets of a topological space. lK 1.2.2. 22: the generic notation for a field (German lKorper). lK[x] 1.3.2. 31: for a field lK (or, more generally, a ring R) and an "indeterminate" x, the set of all polynomials of the form n

L

amx m , n E N, am ElK (or am E R).

m=O

Example 1. Clz] is the vector space of all polynomials in the (complex) variable z and with complex coefficients. Example 2. Z[x] is the ring consisting of all polynomials in the indeterminate x and with coefficients in Z. ker1.1.3. 4 C 1.2.2. 21: Lie algebra; 2.2.1. 104: the set of all Lebesgue measurable subsets of lRn. CeQ 1.1.4. 9: the category of locally compact (topological) groups. LeV 5.2. 229: locally convex vector space. Lip a 2.5.2. 178: generalized Lipschitz condition: for positive a, f is in Lip a at a iff for some positive K (the Lipschitz constant) and some positive 6, Ix - al < 6 => If(x) - f(a)1 :5 Klx - ala. lP 2.3.1. 159 L1(G) 2.4.2. 175 LP (X, C), 1:5 p 2.3.4. 169 Li (0, II) 4.2.1. 220: the set of lR-valued random variables defined on 0 and square integrable with respect to II. ib) 2.2.1. 114, 2.2.1. 123: of the arc 'Y : [0,1] 3 t 1-+ lRn , its length n

L

sup

II'Y (ti) -

'Y (ti-dll, mEN.

o=to<···
L b*) 2.2.1. 114: of the arc-image 'Y*, the infimum of the lengths for all parametric descriptions s. loo 2.3.2. 164 liminf 2.1.3.67: of a sequence {an}nEN of real numbers,

inf sup {am : m ~ n } ;

nEN

of a sequence {An}nEN of sets,

.e ('Y 0

s)

Symbol List

252

i.e., the set of points belonging to all but finitely many An. lim sup 2.1.3.67: of a sequence {an}nEN of real numbers, sup in£{ am : m 2:: n } ; nEN

of a sequence {An}nEN of sets,

i.e., the set of points belonging to infinitely many An.

m 2.3.1. 159: the set of all bounded sequences of complex numbers. mA

1.3.1. 26

min 1.1.2. 4: the integer n is an integral multiple of the integer m. (m, n) 2.1.1. 48: the Greatest Common Divisor (GCD) of the integers m, n. M(m, n) 1.3.3. 38 m mod k 2.1.3. 80: for k and m in !Ii, in [0, k-1) the unique natural number T such that for some natural number q, m = qk + T. Mat mn 1.3.1. 29: the set of m x n matrices with entries from a field K; more generally, MatrA denotes the set of all K-valued functions on r x A. !Ii 1.1.2. 3: the set of natural numbers. NBG 5.2. 228: the axiom system proposed by von Neumann, Bernays and Godel for set theory. N 2.4.1. 173: the set of generalized nilpotents in a Banach algebra. ND 2.3.3. 165: in JRIR , the set of nowhere differentiable functions. Nondiff(f) 2.1.1. 42, 2.1.2. 56 o 4.2. 219: Landau's "little 0" notation: for some a,

f(t) = o(g(t)) <=> lim f((t)) = t-a 9 t

o.

o 1.2.3. 25: in a vector space, the zero vector: x + 0 = x. o 2.2.1. 105: the set of open sets in a topological space X. 1P12.1.2.65: for a partition P ofan interval in JR, the greatest of the lengths of the (finitely many) disjoint subintervals constituting P. P(E) 2.2.1. 105: the power set of E, i.e., the set of all subsets of the set E. pn 2.2.1. 114: the parallelotope [0, l)n. PLPP 1.3.3. 37: the Primal Linear Programming Problem. q(m, n) 1.3.3. 38 Q 1.1.5. 15: for a topological space X, in (JH[*)x the multiplicative group of functions f such that both f and j are bounded and continuous; 1.3.3. 38: a (convex) polyhedron that is the intersection of a finite set of half-spaces in JRn • Q 1.1.4. 6: the set of rational numbers.

Symbol List

253

Q' 2.1.3. 67: the set of limit points of the set Q. QL 1.1.4. 8: quotient lifting. 'R 1.1.5. 16: for a topological space X, in {JR+)x the set of functions f such that both f and :7 are continuous and boundedj 2.2.3. 147: for

a relation R in N ~f {I, ... ,n}, the subset { (i, j) : iRj} of N x Nj 2.4.1. 172: the radical of a (Banach) algebra. R 5.1. 224: the set of rules of inference of a formal logical system. JR 1.1.4. 6: the set ofreal numbers. JR+ 1.1.5. 15: the multiplicative group of positive real numbers. JR(n,+) 1.3.3. 37: the nonnegative orthant

of JRn . sgn 2.1.2.64: the signum function: z E C ~ sgn(z)z = Izi. span(S) 2.3.3. 168: of a subset S in a vector space, the set of all (finite) linear combinations of vectors in S. Sl 2.2.3. 144: in JRn the set {x : IIxll = 1 }. sn 3.3.1. 208: a homeomorphic image of Sl in JRn +!, q.v.j whereas SR is used when the metric properties of the surface of a sphere are discussed, sn is used when the topological properties of the surface of a sphere are studied. S 2.2.1. 103: a u-ring of (measurable) sets. Sn 1.1.2. 3, 1.1.4. 8: the (symmetric) group of all permutations of {I, ... , n}. SD 2.3.3. 166: in JRIR , the set of somewhere differentiable functions. S(A) 1.1.4. 5: the intersection of the set of all u-rings containing the set A of sets, i.e., the u-ring generated by the set A of sets. S(A) 2.1.3. 69: for a sequence A ~f {an}nEN, the formal sum, i.e., without regard to convergence, L~=l an· S,8(A) 2.1.3. 69: for a subset f3 of N, and a sequence A ~f {Bn}nEN the formal sum LnE,8 an· SlI'(A) 2.1.3. 69: for a sequence A ~f {an}nEN and a permutation 7r in the set of II of all permutations of N, the formal sum L~=l all'(n)' SI I(A) 2.1.3. 69: for a sequence A ~f {an}nEN, the formal sum 00

L lanl ($ 00). n=l

SO(n) 2.2.3. 144: over JR, the set of all n x n orthogonal matrices M such that det (M) = 1. SR(f) 2.6.2. 181: on YR, the set of singularities of the function f. supp(f): the support of the function f.

Symbol List

254

T 1.1.5. 18, 2.3.3. 165: the set of complex numbers of absolute value one; equivalently, the set R./Z. TR: the set of complex numbers of absolute value R. T- I 1.3.1. 26: the inverse (if it exists) of the map T. TJ 2.1.2. 65: the total variation of the function f. TJP 2.1.2. 65: for a function f in R.1 and a partition P of the interval I, the total variation of f with respect to the partition P. T rlR 2.3.4. 170: the set of Transcendental R.eal numbers. U 1.2.3. 24: a uniform structure. var 2.3.1. 161: the total variation of a function. V· 1.3.2. 34: the dual space of the vector space V. [V] 1.3.1. 25: the set of endomorphisms of the vector space V. [V]inv 1.3.1. 26: the set of invertible elements in [V]. (V]sing 1.3.1. 26: the set of singular (noninvertible) elements in [V]. [V, W]1.3.1. 29: the set of morphisms (linear maps) of the vector space V into the vector space W. wf 5.1. 224: well-formed formula. WF 5.1. 223: the set of well-formed formulae. (x, y) 1.3.2. 33: the scalar product of the vectors x and y. x ~ y 1.3.3. 37 x !:: y 1.3.3. 37 X-I 1.1.1. 1: the inverse of x. (X, d) 2.1.1. 43: a metric space X with metric d. (X,S,/-I.) 2.2.1. 103: the measure situation consisting of the set X, the (7ring S of (some) subsets of X, and the (countably additive) measure /-I. : S 1-+ [0,00]. X 0 Y: for two sets X and Y in a set S where there is defined a map S x S 3 {x,y} I-+Xoy E S,

the set {x 0 y : x E X, Y E Y}. Example 1. If S is a group, if XES, if T c S, and if a 0 b ~f ab then

=

=

xT def{ xt: t E T } ,Tx def{ tx

t E T}.

Example 2. If Co is the Cantor set in R. then Co + Co ~f {x

+y

: x, y E Co} (= [0,2]).

X Y 1.1.4. 6: the set of all maps of the set Y into the set X. X \ Y 1.1.2. 4, 2.1.1. 44: {x : x E X, x ~ Y}. '1. 1.1.5. 12: the set of integers. '1.2, '1./2 1.3.1. 28: the finite field {O, I}. ZF 5.2. 228: the Zermelo-Fraenkel formulation of the axioms of set theory. ZFC 5.2. 229: ZF together with the Axiom of Choice.

Symbol List

255

ZFCCH 5.2. 229: ZFC together with the Continuum Hypothesis. ZFCGCH 5.2. 229: ZFC together with the Generalized Continuum Hypothesis. ZFS 5.2.229 ~ 1.1.4. 8, 2.2.2. 138: the inclusion map. :!. 1.1.3. 4: the morphism t/J maps to. 'Y 2.2.1. 113 'Y. 2.2.1. 113: for an arc 'Y : 10,1] 3 t ....... 'Y(t) E IR n the set (image) 'Y ([0,1]). r(A) 2.1.3. 69: for a sequence A ~f {an}nEN and the set 4> of all finite subsets t/J of N the set {S",(A) : t/J E 4> }. 6).,1-' 1.3.2. 33: Kronecker's "delta function," i.e.,

= {I

6)., I-'

0

if A = I-' otherwise.

An 2.2.1. 104: Lebesgue measure in IRn. A*, A* 2.2.1. 107: Lebesgue inner resp. Lebesgue outer measure in IRn. p., p. 2.2.2. 138 uT(A) (~f limm-+ oo Um,T(A)) 2.1.3. 75

um,T(A) (~f E~=l t mn 8 n(A)) 2.1.3. 75 u(T) 1.3.2. 31: the spectrum of the morphism T. v « I-' 2.2.2. 137: the measure v is absolutely continuous with respect to the measure 1-', i.e., I-'(A) = 0 => v(A) = o. n 2.1.3. 69: the set of all permutations of N. t/J 2.1.3. 69: a finite subset of N. 4> 2.1.3. 69: the set of all finite subsets of N. XA 1.1.4. 5: the characteristic function of the set A: XA(X)

= {I o

E

if x A otherwise.

w (I, xo, f) 2.3.1. 161: the f-modulus of continuity of f at Xo. w (I, f) 2.3.1. 161: the f-uniform modulus of continuity of f. #(S) 1.1.2. 2: the cardinality of the set S. 8A 2.2.1. 121, 2.6.1. 180: the boundary of the set A in a topological space. => 1.1.1. 1: "implies." ....... 1.1.1. 1: "maps to." {:} 1.1.2. 4: "if and only if' ("iff"). E 1.1.2. 4: "is a member of." C 1.1.3. 4: "is contained in" ("is a subset of'). 1.1.2. 4: the empty set. ~ 1.1.4.6: "approaches," "converges (to);" 1.1.5.17: "maps to" (in (commutative) diagrams); 5.1. 223: "implies" (in formal logic). :J 1.1.4. 8: "contains."

o

Symbol List

256 ~

¥

1.1.5. 10: in group theory, "is isomorphic to," 2.2.3. 145, 3.1.1. 186: in Euclidean space, "is congruent to," 3.3.209: in topology, "is homotopic to." 1.1.5. 10: "is a proper subset of."

! 1.1.5. 17:

"maps to" (in (commutative) diagrams); 2.1.1. 49: "approaches from above," "decreases monotonely (to)." i 2.1.1. 49: "approaches from below," "increases monotonely (to)." S 1.1.5. 18: in topological contexts, the (topological) closure of the set S. o 1.2.2. 21: (binary operation); 2.1.1.42: (composition of functions). Al:JB 1.2.3. 22: the union of the two disjoint sets A and B. 11···11 1.3.1. 26: the norm of the vector···; 2.4.1. 172: the norm of the linear map···. 11/1100 2.1.2.64: for a measure situation (X, S, It) and a measurable function 1 in eX, inf{M: It({x: I/(x)I~M})=O} (~oo).

II/lIp

2.3.1. 159: for a measure situation (X, S, It), p in lR \ {O}, and a measurable function 1 in eX, l.

(Ix I/(x)IP dlt)"

(~oo).

>- 1.3.3. 37: partial order (strictly greater than). t 1.3.3. 37, 2.3.4. 169: partial order (greater than or equal to). ~ 2.1.2. 53: "converges uniformly (to)." ~. 2.1.3. 86, 2.2.2. 134: "converges almost everywhere (to)."

V 2.2.1. 122 d~ 2.2.2. 135: "converges dominatedly (to)." m~as 2.2.2. 134: "converges in measure (to)." IIjP 2.2.2. 134: "converges in p-norm (to)." 2.3.4. 169: supremum (of a pair); 5.1. 223: logical "or." A 2.3.4. 169: infimum (of a pair); 5.1. 223: logical "and." -, 5.1. 223: logical "not." V 5.1. 223: logical "for all." 3 5.1. 223: logical "there exist(s)." l:J 1.1.2. 4, 2.1.2. 63, 2.2.1. 149: used instead of Uto signify the union of a set of pairwise disjoint sets. V

GLOSSARY /INDEX

The notation a.b.c. d indicates Chapter a, Section b, Subsection c, page d; similarly a.b. c indicates Chapter a, Section b, page c.

Abel, N. H. 2.1.3. 76 Abel summable 2.1.3. 76 Abel summation 2.1.3. 76 abelian 1.1.4. 8: of a group G, that the group operation is commutative. abelianization 1.1.5. 15: for a group G and its commutator subgroup Q(G), the (abelian) quotient group G/Q(G). absolutely continuous 2.1.2. 55, 2.1.2. 65, 2.1.3. 87: of a function f in CIR , that!, exists a.e. on [a, b) and that for x in [a, b),

f(x)

= f(a) +

1 x

f'(t) dt

cf. Exercise 2.1.2.15. 65; 2.2.2. 137: of a measure v with respect to a measure p., that every null set (p.) is also a null set (v). - convergent 2.1.3. 69: of a series E~=l an, that E~=llanl < 00. ADIAN, S. I. 1.1.5. 11

adjacent 1.1.5. 10: in the context of free groups, of two words Wi and W2, that there is an x such that for some words u and v, Wi = UXEX-EV and W2 = uv. adjoint 1.3.1. 25: of a matrix (aij)7,'j~l' the matrix (bij)~j':::i in which bij = aji; 1.3.2 34: of a linear transformation T : V ....... W between vector spaces, the linear transformation T* : W* ....... V* between their duals and satisfying w* (Tv) = T*w*(v). ALEXANDER, J. W. 3.2.2. 206 Alexander's horned sphere 3.2.2. 206 algebra 1.2.2. 21, 2.3.4. 169, 2.4.1. 172: a ring R that is a vector space over a field K and such that if a E K and x, y E R then

a(xy) = (ax)y = x(ay). algorithm 5.1. 227: a (computer) program for mapping Z into itself. (The preceding definition is a colloquial version of Church's thesis.) almost every section, point, etc. 2.2.1. 110: every section outside a set of sections indexed by a null set, every point outside a null set, etc. almost everywhere (a.e.): in the context of a measure situation, "except on a set of measure zero (a null set)." 257

Glossary/Index

258

alternating group 1.1.4. 8, 1.1.6. 19: the group An of even permutations of the set {I, 2, ... ,n}. - series theorem 2.1.3.79: If an E JR, an = (-I)nla n l, and lanl ! 0 then E:'=l an converges. alternative (division) algebra 1.2.2. 22: an algebra in which multiplication is neither necessarily commutative nor necessarily associative. analytic continuation 2.6.2. 180: for a region n* properly containing a region n in which a function I (in Cc ) is analytic, the process of defining,a function analytic in n* and equal to I in n. - function 1.3.1. 26, 2.1.1. 51: a function I in CC and such that f' exists (in some region n). - set 5.1. 228: the continuous image of a Borel set. antiderivative of order k 2.1.2. 62: for a function I, a function F such that

r

F(k)

= I.

L. 3.2.2. 207 Antoine's necklace 3.2.2. 207 ApPEL, K. 3.1.2. 197 arc 2.2.1. 113: a continuous map of'Y : [0,1]1-+ X of [0, I] into a topological space X. ARCHIMEDES 1.2.3. 23 Archimedean 1.2.3. 23: of an ordered field K that if p, q E K and 0 < p < q then, for some n in N, q < np. arc-image 2.2.1. 113: the range of an arc. arc wise connected 1.3.2. 37: of a set S in a topological space X, that any two points of S are the endpoints of an arc in S. area 2.1.2. 58: of a subset S of JR2, the value of the Riemann (or, more generally, the Lebesgue) integral ANTOINE,

/ L2 Xs(x, y) dA2(X, y) (if it exists). The "problem of 'surface' area" for (images of) maps from JRm to JR n , especially when m < n, is difficult. One of the difficulties, when 2 = m < n = 3, is discernible from a reading of the discussion of Example 2.2.1.11. 118, Remark 2.2.1.4. 121, Note 2.2.1.4. 121, and Exercise 2.2.1.12. 123. For extensive discussions of the topic cf. [01, 02, Smi]. ARTIN, E. 3.2.2. 206 ARZELA, C. 2.3.1 162 ASCOLI, G. 2.3.1. 162 Ascoli-Arzela theorem 2.3.1. 162: A uniformly bounded set of equicontinuous JR-valued functions defined on a compact metric space contains a uniformly convergent subsequence. associativity 1.1.1. 1: of a binary operation, that always a(bc) = (ab)c. auteomorphism 3.1.2. 196: a homeomorphism of a topological space onto itself.

Glossary jIndex

259

autojection 2.2.1. 114: a bijection of a set onto itself. automorphism 1.2.2. 22: a bijective endomorphism (whence an "autojective" endomorphism). average 2.1.3. 74: of a finite set {8l,' .. ,8n } of numbers, the number 81

+ ... + 8 n n

axiom 1.1.1. 1, 5.1. 224 Axiom of Choice 1.1.4. 6, 2.3.3. 167, 5.2. 229: If {A~hEA is a set of sets, there is a set A consisting of precisely one element from each A~. - - Solovay 5.2. 229

Baire, R. 2.1.1. 43 Baire's (category) theorem 2.1.1. 43: The intersection of a countable set of dense open subsets of a complete metric space is a dense G6. ball: see closed ball, open ball. BANACH, S. 1.3.2. 37, 2.1.1. 51, 2.2.3. 145, 2.3.1. 156, 2.4.1. 172 Banach algebra 1.3.2. 37, 2.4.1. 172: an algebra A (over lR or q that is a Banach space and such that for any scalar a and any vectors x and y the relations lIaxYIl = lallixyll ~ lalllxllilyll obtain. - space 1.3.2. 31, 2.1.1. 51, 2.3.1. 156: a complete normed vector space over lR or C. - -Tarski paradox 2.2.3. 145 base 2.2.1. 112: for the topology of a space, a set S of open sets such that each open set of the topology is a union of sets in S. basic neighborhood (in a Cartesian product) 2.2.1. 113: for a point X

def {

=

x~

}

~EA

in a Cartesian product n~EAX~ of topological spaces, a set that is a Cartesian product in which finitely many factors, say those corresponding to the finite set {AI. ... , An}, are neighborhoods U~i of the components X~j' 1 ~ i ~ n, and in which the remaining factors are the full spaces X~I, A' ¢ {AI,"" An}. - variables 1.3.3. 39: in linear programming, the variables constituting the complement of the set of free variables, q.v. basis 2.3.1. 156: in a topological vector space, a Schauder basis. BAUMSLAG, G. 1.1.5. 11 BESICOVITCH, A. S. 2.2.1. 118, 2.2.1. 123 BERNAYS, P. 5.2. 229 BESSEL, F. W. 2.3.1. 161 Bessel's inequality 2.3.1. 161: for an orthonormal system {X~hEA and any vector x in a Hilbert space, the relation: E~EA l(x,x~)12 ~ IIx1l2.

Glossary/Index

260

between (vectors u and w) 2.3.3. 167: of a vector v, that it lies on the convex hull of u and w. betweenness 3.1.1. 186: used in the axiomatic foundation of geometry. bicontinuous 2.1.1. 42: of a bijection, that it and its inverse are continuous. BIEBERBACH, L. 2.6.6. 184 Bieberbach conjecture 2.6.6. 184 bifurcation (superbifurcation) 2.5.2. 179: the failure of a differential equation to have a unique solution at some point. bijection 1.1.5. 10, 2.1.3. 69: an injective surjection, i.e., a one-one map b: X H Y such that b(X) = Y. bijective 1.1.5. 9: of a map, that it is a bijection. binary marker 2.1.1. 52 -representation 1.1.4. 7 biorthogonal 2.3.1. 159: of two sets {X~hEA and {XU~EA of vectors, the first in a vector space V and the second in the dual space V·, that

bisection-expansion 2.2.1. 127 BLAND, R. G. 1.3.3.40 Bland's algorithm 1.3.3. 40 BOCHNER, S. 4.1.1. 211 BOLZANO B. 3.2.2. 200 Bolzano-WeierstraB theorem 3.2.2. 200: A bounded infinite subset of a compact metric space X has a limit point (in X). BooLOs, G. 5.1. 228 BOONE, W. W. 1.1.5. 11 BOREL, E. 2.2.1. 105, 4.2. 221 Borel measure 4.2. 221: of a measure p" that its domain of definition is the set of Borel sets in a topological space X. - set 2.2.1. 105: in a topological space X, a member of the a-ring generated by the set of open sets of X. boundary 3.1.2. 195: of a set S in a topological space X, the set as consisting of the points in the closure of both S and of X \ S:

asd,~,fsnX\S. bounded 2.4.1. 172: of a homomorphism T : A spaces, that

IITII ~f sup { IIT(x)1I T is bounded iff T is continuous.

IIxll =

H

1}

B between normed

< 00;

Glossary/Index -

261

approximation property 2.3.1. 158: of a normed vector space V, that there is in [V] a sequence {Fn}nEN such that sUPnEN IlFnll < 00 and for every vector x, lim IIx - Fnxll = O. n ..... oo

- variation 2.1.2. 54, 2.1.3. 88, 3.1.2. 192: of a function

f in R.[a,bl, that

n

sup

a~:r;l <···<:r;n~b

L If(Xk) - f(Xk-ll <

00,

n E N.

k=2

BRANGES, L. DE 2.6.6. 184 bridging functions 2.1.2. 62, 2.2.2. 139 BRIESKORN, E. V. 3.3. 208 BRITTON, J. L. 1.1.5. 11 BROUWER, L. E. J. 3.1.2. 195, 3.1.2. 197 Brouwer's invariance of domain theorem 3.1.2. 197: If!l is an open subset of R. 2 and if f : !l t-+ f(!l) C R. 2 is a homeomorphism, then f(!l) is open. BURNSIDE, W. 1.1.5. 11 Burnside question 1.1.5. 11

canonical relation 2.2.3. 147 CANTOR, G. 1.1.4. 7, 2.1.2. 55, 2.1.3. 85 Cantor function 2.1.2. 55 - -Lebesgue theorem 2.1.3. 85 - set 1.1.4. 7 -'s theorem 5.1. 225 cardinality 1.1.2. 2: of a set, the bijection-equivalence class of a setj alternatively, the least ordinal among all bijectively equivalent ordinals. CARLESON, L. 2.3.1. 158 Cartesian product 2.2.1. 112: of an indexed set {X~hEA' the set of all "vectors" {X~hEA such that X~ E X~. category 1.1.4. 8: a complex consisting of a) a class C of objects, A, B, . .. j b) the class of pairwise disjoint sets [A, B] (one for each pair A, B in C x C) of morphismsj c) an associative law 0:

[A,B] x [B,C]

3 (f,g)

t-+

go f E [A,C] ~f [A,B]

0

[B,C]j

of composition of morphismsj d) for each A in C a morphism lA in [A, A] and such that if f E [A, B] and 9 E [C, A] then f 0 lA = f and lA 0 9 = g. Example 1. The class of all groups is a category g if [A, B] is the set of homomorphisms ifJ : A t-+ B, if ifJ 0 t/J denotes the composition of ifJ and t/J, and lA is the identity map. Example 2. A partially ordered set {S, -<} is a category S if the objects are the

262

Glossary/Index

elements A, B, ... of S and if the set of morphisms associated with the pair (A, B) is

[A, B] ~f

{

if A j B ~ A -< B V A = B otherwise.

0(A,B)

Since j is transitive, i.e.,

A j B AB j G

~

A j G,

the composition of morphisms is well-defined:

[A,B]

0

[B,G] = {[A,C]

o

if [A,~] # 0, [B,G] otherwIse.

#0

Since j is reflexive, i.e., A j A, for each A in S, the identity lA exists and is [A, A]. In particular, the set of morphisms need not be a set of maps. (See also first category and second category.) CAUCHY, A. L. DE 1.2.3. 24 Cauchy complete 1.2.3. 24: of a uniform space X, that every Cauchy net in X has a limit in X. - completion 1.2.3. 24: for a uniform space X, the set of equivalence classes of Cauchy nets. - criterion 2.1.3. 71: for a complete uniform space X, the statement that a net converges iff it is a Cauchy net. - -Kowalewski theorem 2.5.2. 179: For a function f in aRn and analytic in a neighborhood of 0, let the partial differential equation

have coefficients analytic near 0 and let the coefficient of

be different from zero at O. Then if ¢ is analytic near 0 the differential equation has a unique solution u, analytic near 0 and satisfying

(- 0)kl ... (0OXI

when

[Ho].

)kn (u-¢)=O

oXn

Xl

= ... = Xn = 0 and al + ... + an < N

Glossary/Index

263

net 1.2.3. 24: in a uniform space X with uniformity U, a net {X~hEA such that for each U in U there is a AO such that (x~, X~/) E U if A, A' tAO' - product 2.1.3. 82: of two series E::'=o a" and E:=o bm , the series -

A. 1.2.2. 22 Cayley algebra 1.2.2. 22 central limit theorem 4.1. 214: If {f"}"EN is a sequence of independent random variables such that CAYLEY,

1x1,,(W)dw = 0

Ix

1,,(w)2 dw

and such that for every positive

=1

f,

lim _1_ f l(w)2 dw .,fii J{w : Ifn(w)I~€v'n}

= 0,

" ..... 00

then · def li 11m g" = m

" ..... 00

It + ... + I" r.;: yn

" ..... 00

def

=9

exists almost everywhere and

P{w : g(w):5 x} =

1 .tn=

y27r

1 z

2

t exp(--)dt.

-00

2

E. 2.1.3. 76 Cesaro summation 2.1.3. 76 chain 2.5.2.179: a sequence {FJ} of functions, indexed by and defined over intervals J of a partition of an interval [a, b), and such that the FJ and all their derivatives coincide at common endpoints of the intervals J. characteristic function 1.1.4. 5, 2.1.1. 46, 2.2.1. 104, 2.2.2. 133: of a set A, the function XA such that CESARO,

XA ( x ) CHARNES, CHURCH,

A. 1.3.3.40 A. 5.1. 227

def{1

= o

ifxEA if otherwise.

264

Glossary/Index

circle of convergence 2.6.2. 181: for a power series ~:=o anzn , in C the set

{z :

Izl < R}

in which the series converges absolutely and outside the closure of which the series diverges. class 5.2. 229 closed 2.1.1. 43: of an interval in JR, that it contains its endpoints; of a set A in a topological space, that A is the complement of an open set; 2.2.1. 113: of an arc that its endpoints are equal; 3.2.2. 203: of a map f : X 1-+ Y between topological spaces, that the images of closed sets are closed. - ball 3.2.1. 200: in a metric space (X,d), for a in X, a set of the form

{x : d(x, a) $ r} ; by abuse of language, ball. convex hull 2.3.2. 163: for a set A in a topological vector space, the intersection of the set of all closed convex sets containing A. - graph theorem 1.3.2. 34, 2.4.1. 173: If T : X 1-+ Y is a linear map between Banach spaces and if the graph { (x, Tx) : x EX} is closed in the Cartesian product X x Y then T is continuous. - upper half-plane 3.2.2. 201: {(x, y) : (x, y) E JR2, Y ;::: 0 }. closure 1.1.5. 18: of a set A in a topological space X, the intersection of the set of all closed sets containing A. coaxial 3.1.1. 187: of a set of lines in Euclidean space, that they have a point in common or that they are parallel. codimension 2.3.2.165: of a subspace W of a vector space V, the dimension of the quotient space V /W. coefficient functionals 2.3.1. 159: for a (Schauder) basis {bn}nEN of a Banach space W, the maps b~ : W :3 x 1-+ b~(x) E C such that for all x in W, x = ~:=l b~(x)bn' COHEN, P. 5.2. 229 column vector 1.3.1. 26: an n x 1 matrix. commutative 1.1.5. 16: of a binary operation 0 that always x 0 y = y 0 x. - diagram 1.1.5. 17: a figure in which various arrows (-, -, 1, !) signify mappings (morphisms) and in which any composition of mappings starting at one entry E in the figure and ending at another entry E' of the figure is a mapping depending only on E and E'. commutator 1.1.5. 15: of elements a and b in a group, the element aba-1b- 1. - subgroup 1.1.5. 15: in a group G, the minimal (normal) subgroup Q such that G/Q is abelian; alternatively, the subgroup generated by the set of all commutators. commute 1.1.5. 13: of two elements a and b, and with respect to a binary operation, that ab = ba. -

Glossary jlndex

265

in pairs 1.3.1. 27: of a set S of elements, and with respect to a binary operation, that any pair in S commutes. compact 1.1.4. 5, 2.1.1. 42: of a topological space, that every open covering contains a finite subcovering. -neighborhood (of a point P) 1.1.4. 9: a compact set that is a neighborhood of P. comparison test 2.1.3. 72 compatible 2.2.3. 147 complete 1.2.3. 23: of an ordered field lK, that each set bounded above has a least upper bound. completely regular 1.1.5. 12: of a topological space X, that if F is closed in X and x ~ F then there is in C(X, IR) an f such that -

f(y)

1 ify=x if y E F.

={ 0

completeness (of an orthonormal set) 2.1.3. 86 complete orthonormal set 2.1.3. 86, 2.3.1. 156, 2.3.4. 169: in a Hilbert space, an orthonormal set S for which S.L = {OJ. complex dimension 3.3. 208: of a vector space over C, the cardinality of any (hence every) C-basis. component 1.1.5. 14: of a vector {X"heA, some x,,; 2.2.1. 115: in a topological space, a maximal connected subset. composition 2.1.1. 42: of maps g : Y ~ Z and f : X ~ Y, the map go f : X :3 x ~ g (f(x» E Z. concentrated 2.2.2. 138: For (X,S,J.L), J.L is concentrated on A in S iffor all E in S, J.L(A n E) = J.L(E). conditional 2.3.1. 160: of a (Schauder) basis in a Banach space, that the series representing some element of the space fails to converge unconditionally. conditionally convergent 2.1.3. 69: of a series in a Banach space, that some permutation of its terms yields a divergent series; alternatively, that for some distribution of signs on the terms of the series, divergence results. conformally equivalent 2.6.3. 183: of two regions in C, that there is for each a univalent biholomorphic map of it onto the other. congruence 2.2.3. 147: between two sets A and B in IR3 , a relation ~ specifying that there is a rigid motion T such that T(A) = B, 3.1.1. 187: an axiomatized concept in (Euclidean) geometry. congruent 2.2.3. 145: of two sets in IRn , that there is a rigid motion, i.e., the composition of a finite set of transformations drawn from the set of translations and proper orthogonal linear transformations, mapping one set onto the other; 3.1.1. 186: an axiomatized relation in Hilbert's presentation of Euclidean geometry. - matrix 1.3.2. 35: for a matrix A, a matrix B such that for some invertible matrix T, B = T* AT.

Glossary jlndex

266

conjugate 1.1.5. 13: of a quaternion q ~f al + bi + cj + elk, the quaternion _def , elk . q=a 1 - b'I-CJ- linear 1.3.2. 33: of a map T : V 1-+ C of a vector space V over C, that a E C => T(ax) = aT(x). - symmetric 1.3.2. 33 connected 2.1.1. 43: of a topological space X, that it is not the union of two nonempty and disjoint open sets. consequence 5.1. 224 constant (in formal logic) 5.1. 223 contain a constant 2.2.3. 151: of a relation R on {I, ... ,n}, that for some k, (iRk), 1 ~ i ~ n; by abuse of language, R contains the constant k. content: see n-dimensional content. continuous 1.1.4. 5, 2.1.1. 42, 3.2.2. 203 Continuum Hypothesis 5.2. 229: The cardinality of the set S of all wellordered finite or count ably infinite sets no two of which are in orderpreserving bijective correspondence is the cardinality of 1R: Nl = 2No. contradictory 5.1. 224: of two wfs, that each is the negation of the other. convergent (converges) 1.2.3. 24: A net in a topological space is convergent (converges) iff the net has a limit. converges absolutely 2.1.3. 82: of a series E:'=1 an consisting of real or complex terms, that E::'=llanl < 00. - uniformly 2.1.2.56,2.1.3.84: of a sequence {fn}:'=o of functions defined on a set S that for each positive f there is an N(f) such that for all x in S, Ifn(x) - fo(x)1 < f if n > N(f). convex 1.1.4. 7: for an open set U in IR and of a function f on lR u , that if t E [0,1] and x, y, tx+(l-t)y E U then f(tx+(l-t)y) ~ tf(x)+(l-t)f(y): "the curve lies below the chord;" 2.3.3. 167: of a set S in a vector space V, that (u, v E S) " (t E [0,1]) => tu + (1 - t)v E S. - hull 2.3.3. 168: of a set S in a vector space, the intersection of the set of all convex sets containing S. - polyhedron 1.3.3. 38: in a vector space, the intersection of a finite number of half-spaces. convolution 1.1.4 5, 2.1.3. 74, 2.3.4. 170, 2.4.2. 176: of two functions f and g defined on a locally compact group G (with Haar measure IJ.) and in Ll (G,q, the function

f corner 4.2. 216: for

*g : G 3

(Xl, . .•

x

1-+

fa

f(t- 1 x)g(t) dlJ.(t).

,xn ) in IRn , ,the set

coset 1.1.2. 3: of a subgroup H of a group G, for some x in G, a set of the form xH or Hx.

Glossary /Index

267

countable 1.1.4. 5: of a set, that its cardinality is that of N. count ably additive 2.2.1. 104: of a C-valued set function C), that if {An}nEN is a sequence of pairwise disjoint sets in the domain of C) then 00

C)

(l:JnENAn) =

:E

C)

(An) .

n=l

- subadditive, (see subadditive): of a nonnegative set function

C),

that

counting measure 2.2.2. 134, 2.2.2. 141: for a measure situation (X, 2x , 1'), the measure I' such that for every subset A of X, I'(A)

= { #(A) 00

if A is ~nite otherwIse.

cycling 1.3.3. 38: in linear programming by the simplex algorithm, the phenomenOn in which a finite set of vertices is recurrently visited without the conclusion that anyone of them is optimal: the algorithm cycles. cylinder set 4.2. 216: in a Cartesian product, a set determined by conditions on finitely many vector components; 4.2. 217: in a vector space V, a set Zxr ,... ,x:,;A defined by a finite subset {xi, ... , x~} of the dual space V* and a Borel subset A of IRn :

Zxr, ... ,x:';A ~f {x : x E V, (xi (x), ... ,x~ (x» E A}. (C, a)-summable 2.1.3. 76

Daniell, P. J. 2.2.1. 104 Daniell integral 2.2.1. 104: a linear functional I defined On a linear lattice L of extended lR-valued functions and such that:

ItO =* 1(1) ~ 0; In ! 0 =* I (In) ! o.

I

EL"

G. 1.3.3.38 M. 2.3.1. 157 decreasing 1.2.3. 24: for ordered sets (Xi, b), i = 1,2, and of an I in yX, that Xl b YI =* I (Xl) b I (yd· DEDEKIND, R. 5.1. 224 dense 1.1.4. 5, 2.1.1. 48: of a subset A of a topological space X, that the closure A of A is X; equivalently, A meets every nonempty open subset ofX. DANTZIG, DAVIE, A.

Glossary jlndex

268

derivation 2.4.1. 173: in an algebra A, a linear endomorphism D such that for x, yEA, D(xy) = D(x)y + xD(y). derived set 3.2.2. 200: the set of limit points of a set. DESARGUE, G. 3.1.1. 187 Desargue's theorem 3.1.1. 187 diagonable 1.3.1. 26: of a SQUARE matrix A, that there is an invertible matrix P such that p-l AP is a diagonal matrix. diagonal (matrix) 1.3.1. 26: a SQUARE matrix (aij)~j~l such that aij = 0 if i ~ j. diameter 2.2.1. 126: of a set S in a metric space (X, d), sUPZ,IIES d(x, y). DIAMOND, H. 2.2.1. 110 diffeomorphism 3.3. 208: a Coo surjective homeomorphism D : X 1-+ Y between differential geometric structures X and Y. difference set 2.2.1. 109: for two subsets A, B of a group resp. abelian group, the set {ab- 1

:

a E A, b E B} resp. {a - b : a E A, b E B} .

differentiable (at xo) 1.1.4. 7: of a vector function f

~f (/I, ... ,In) : am :3 x ~f

(

7)

1-+

(/I~X») In (x)

Xm

that there is in [am, an] a T such that

II (f (xo + b) -

lim

f (xo) - T (xo) b)

IIhll

h;60,lIhll-O

II = o.

The vector T (xo) is the derivative of f at Xo. If f is differentiable at each point of the domain R of f then f is differentiable on R. When, for a choice of bases, X for am and Y for an, T is realized as an m x n matrix

then Txy is the Jacobian matrix 8f(x)

~

I

I

def 8(/I,···,ln) X=Xo = 8 (Xl! ... , Xm) Zl=Zlo, ... ,Zm=Zmo·

= n then det (Txy) is the Jacobian determinant for f. If m = = 1 and X = Y = (1) and fresp. Xo is written I resp. Xo then

If m n

Txy (xo)

dl(x) I = -dX

Z=Zo

= det [Txy (xo)].

Glossary /Index

269

differential geometric structure 3.3. 208: a Hausdorff space X, an open covering U of X, and a set ~f {cPu }UEU of homeomorphisms

cPu: U 1-+ JRn such that if U n U' ~f W

cPu' 0 cPc/

E

t- 0 then

Coo (cPu(U) n cPu' (U'), cPu(U) n cPu' (U')) .

The structure is an n-dimensional differentiable manifold. U. 2.1.3. 93 Dini's theorem 2.1.3. 93 DIOPHANTUS 5.1. 228 Diophantine 5.1. 228: of a set of polynomial equations, that their coefficients are in Z and that their solutions are to be sought in Z. directed 1.2.3. 24: of a set, that it is partially ordered and that every pair in the set has an upper bound. direct product 1.1.3. 5: for a set of algebraic structures, their Cartesian product endowed with component-wise operations. discontinuity 1.1.4. 7: for a map of a topological space into a topological space, a point where the map is not continuous. discrete topology 2.3.2. 163, 3.2.2. 200: the topology in which every set is open. diset 1.2.3. 24: a directed set. distribution 2.5.2. 179: for the set W ~f COO (JRn , JR) endowed with a suitable locally convex topology, a continuous linear functional on W, i.e., an element of the dual space W* of W. - function 4.2. 216: for a set {/k}~=l of random variables, the function DINI,

divergent 2.1.3. 70: of a series, that it fails to converge. division algebra (division ring) 1.1.5. 13, 1.2.1. 19: an algebraic object governed by all the axioms for a field, save the axiom of commutativity for multiplication. domain 2.1.2.55,2.4.1. 174: for a map T: X 1-+ Y, the set X. dominate 2.1.3. 72 - absolutely 2.1.3. 72 dual space 1.3.2. 34, 2.3.1. 159, 2.3.2. 163: for a (topological) vector space V over a (topological) field K, the set V* of (continuous) linear maps of V into K. dyadic space 2.2.1. 112

edge 1.3.3. 38, 2.2.1. 123: of a polyhedron II in JRn , the intersection of II with n - 1 of the hyperplanes determining II.

270

Glossary/Index

effectively computable 5.1. 227: of an element f in NN, that there is a computer program that, for each n in N, can calculate f(n) in finitely many steps. EGOROFF, D. F. 2.2.2. 133 Egoroff's theorem 2.2.2. 133 eigenvalue 1.3.1. 25, 1.3.2. 31: of an endomorphism T of a vector space V, a number A such that for some nonzero x in V, Tx = Ax. - problem 1.3.1. 25 eligible 1.3.3. 41: of free and basic variables in a PLPP, that they are candidates for SWITCHing. embedding 3.2.2. 206: a homeomorphism ~ : X ...... Y of a topological space X into a topological space Y; in particular, a homeomorphism ~: X ...... JRn . endomorphism 1.3.1. 25: a morphism of an object into itself. endpoints 2.2.1. 113: of an arc or of an arc-image, the images of 0 and 1. ENFLO, P. 2.3.1. 157, 158 entire 2.6.4. 183: of a function f in Ce , that f' exists everywhere in C, Le., f is holomorphic in C. epimorphism 1.1.5. 11: a morphism of an object onto an object. €(t)-channel 2.5.2. 179: in JR2, for a positive function € : JR :3 t ...... €(t) and a continuous function f : JR :3 t ...... f (t) the set { (t, y) : t E JR, f(t) < y < f(t)

+ €(t)}.

€-pad 3.1.2. 197: for a set S in a metric space (X, d) and a positive €,

u {y : d(y,x)

~ €}.

xES

equality 5.2. 229 - modulo null sets 4.1. 212: of two functions, that they are equal almost everywhere. equicontinuous, (see Ascoli-Arzeld theorem): of a set {f~hEA of functions mapping a uniform space X into a uniform space Y, that for V in the uniformity V for Y there is in the uniformity U for X a U such that (x,x') E U

'* (f~(x),b(x')) E V,

A E A.

equivalence class 1.1.2. 3: for an equivalence relation R on a set S, for some a in S, a set of the form {x : xRa}; 2.3.2. 163: in V(X, C), f Rg iff f - 9 = 0 a.e .. - by complementation 2.2.3. 151 - by transitivity 2.2.3. 151 - relation 1.1.2. 3: on a set S a relation R that is reflexive, symmetric, and transitive.

Glossary/Index

271

equivalent Cauchy nets 1.2.3. 24 -norms 2.3.2. 164: on a vector space V, norms constants k and k' and all x in V

II II

and

1111' such that for

IIxll :5 k'lIxll' and IIxll' :5 kllxll· - words 1.1.5. 10 essential singularity 2.1.2. 61: for a holomorphic function f, a singular point that is not a pole and is a limit point of points of holomorphy. EUCLID 1.3.1. 26, 1.3.2. 33 Euclidean 1.3.2. 33: of a vector space, that it is endowed with a positive definite, conjugate symmetric, and conjugate bilinear inner product. - norm 1.3.1. 26: a norm derived from an inner product for a Euclidean vector space. EULER, L. 2.1.3. 86 Euler formula 2.1.3. 86: eit = cos t + i sin t. evaluation map 4.2. 217: for a function space S in some yX and an x in X, a map S 3 f 1--+ f(x) E y. even 1.1.4. 8: of a permutation 11", that nl
nl~i<j~n

(X

1r (i)

-

Xj)

_

1.

x,..(;)) - ,

2.1.3.87: of a map f : V 1--+ W between vector spaces, that for all x in V, f(x) = f( -x). event 4.1. 210 exact 1.1.3. 5: of a sequence G ~ H:J!.. K that the kernel of the morphism t/J is the same as the image of the morphism 4>: ker(t/J) = im(4)). existence/uniqueness theorem (for differential equations) 2.5.2. 177 exotica (in differential topology) 3.3. 208: literally, "outsiders," figuratively, unexpected phenomena in differential topology. expectation 4.1. 210 expected value 4.1. 210 expression 5.1. 223 extreme point 1.3.3. 38, 2.3.2. 162, 2.3.2. 163: in a convex set K, a vector x in K and not, for any y and z in K \ {x} and any t in (0,1), of the form ty + (1- t)z.

factor (of a word) 1.1.5.9 FElT, W. 1.1.6. 18 FEJER, L. 2.1.3. 74, 2.1.3. 75, 2.1.3. 89, 2.3.3. 165 Fejer's kernel 2.1.3. 74

Glossary/Index

272

- -Lebesgue theorem 2.1.3.89 - theorem 2.1.3. 75, 2.3.3. 165: If I E C([-7I", 71"1, C) and 1(-71") = 1(71") then FN * I ~ I on [-71",71"1· field 1.2.1. 19: a set ]I{ in which there are defined two binary operations: . : ]I{

(a, b) 1-+ a . b ~f ab E x ]I{ E (a, b) 1-+ a + b ElK.

x

+ : ]I{ Furthermore both . and

]I{ 3

]I{

+ are commutative and associative and a(b + c)

= ab + ac.

With respect to +, ]I{ is a group (with identity 0) and with respect to ., ]I{ \ {O} is a group (with identity 1). filtering 5.2. 229: of a set 'P ~f {P~hEA of seminorms in a vector space V, that the family B ~f {C~hEA of sets C~ ~f {x : p~(x) < I} is a filter base, i.e., a) B t- 0, 0 rt B, b) if F, G E B then there is in Ban H contained in F n G; if'P is a filtering set of seminorms, the set 0 of (arbitrary) unions of translates of finite intersections of elements of B is a topology for V; if n~EA C~ = 0 then 0 is a Hausdorff topology. finite decomposition 2.2.3. 147, 2.2.3. 150: for a set S, a finite set of nonempty subsets AI' ... ' An such that S = l:J~=1 All:. - intersection property 2.1.1. 45: of a set S ~f {F~hEA of sets, that if n~EA F~ = 0 then for some finite set {AI, ... , An}, n:=l F~t = 0. - rank 2.3.1. 158: of a vector space homomorphism T, that the dimension of the image of T is finite. finitely generated 1.1.5. 11: of a group, that it is generated by a finite set of elements. - presented 1.1.5. 11 first axiom of count ability 3.2.2. 203: satisfied by a topology if, for each point x, there is a countable set S of x-neighborhoods such that contained in every x-neighborhood is an x-neighborhood drawn from S. - category 2.1.1. 43: of a set in a topological space, that it is the countable union of nowhere dense sets. - -order system 5.1. 223: A first-order formal system of logic is specified as follows. 1. The (countably many) symbols of the system are: a. the connectives .., ("negation"), -+ ("implies"); b. the quantifier V ("for all"); c. derived connectives V ("or"), " ("and"), and the derived quantifier 3 ("there exists"); d. predicate letters A, ... , function letters I, ... , variables x, ... , and constants a, ... .

Glossary /Index

273

2. An expression is a finite sequence of symbols. An expression E is a term of the system iff E is: a. a variable, a constantj or b. for terms tll . .. , tn and a function letter f, f (tll ... , tn). 3. If A is a predicate letter and t ll . .. ,tn are terms, A (tll .. . , tn) is an atomic formula. An expression is a well-formed formula (wf) iff is: a. an atomic formulaj or b. ...,A for some wf Ajor c. A -+ B for wfs A and Bjor d. for some variable y and wf A, '1yA. 4. The axioms of the system deal with the relations among wfs A, ... , and are: a. A -+ (B -+ A)j b. (A -+ (B -+ C)) -+ ((A -+ B) -+ (A -+ C))j c. (...,B -+ ...,A) -+ ((...,B -+ A) -+ B)j d. if t is a term free for x in A(x), i.e., in t there is no occurrence of ('Ix) or (3x) then: ('1x)A(x) -+ A(t)j e. if x is not free in A, i.e., every occurrence of x is bound by a quantifier then: ('Ix) or (3x): ('Ix) (A -+ B) -+ (A -+ ('Ix) B)j 4. the rules of inference are: a. modus ponens: B follows from A and A -+ Bj b. generalization: ('Ix) A follows from A. 5. Proper axioms, e.g., the axioms of group theory.

e

e

The descriptor first-order indicates that the system does not deal with predicates or functions as arguments of predicates nor as arguments of quantifiers. The items 1.-5. permit the chaining together of axioms to form proofs of theorems. FISCHER, E. 2.3.4. 169 fixed point 2.2.3. 147: for a relation R, a k such that kRkj 2.2.3. 147: of a rotation U (U ¥- J) of IR3 , a point P such that U(P) = Pj more generally, of a self-map f : X t-+ X, a point x such that f(x) = x. forcing 5.2. 229 formal system of logic 5.1. 223 four-color problem 3.1.2. 198 FOURIER, J. B. J. 2.1.3. 84 Fourier series 2.1.3. 84: a trigonometric series ~nEZ Cn such that for some f Lebesgue integrable on [-11',11'],

v1i-

Cn

=

1 11"

-11'

e-in6

f(O) rn= dO, n E Z. V 211'

Glossary jlndex

274

-

transform 2.1.3. 88: for an / Lebesgue integrable on JR, the function

in

/• : JR 3 tt--+ -1/(x)e-''t x dx . ..fi;JR

Fox, R. 3.2.2. 206 FRAENKEL, FRANKLIN,

A. 5.2. 228 P. 2.3.1. 157,2.3.1. 160

Franklin (functions) 2.3.1. 157, 2.3.1. 160: functions constituting an orthonormal (Schauder) basis for C ([0,1], JR). free abelian group 1.1.5. 16 - group on X 1.1.5. 10 - subgroup 1.1.5. 14: in a group G, a subgroup that is a free group. - subset (of a group) 1.1.5. 11 - topological group on X 1.1.5. 12 - variables 1.3.3. 39: in linear programming, a minimal set of variables in terms of which all variables are linear functions. FREEDMAN, M. H. 3.3. 209 FUBINI, G. 2.1.4. 101, 2.2.1. 130, 2.2.2. 140 Fubini's theorem 2.1.4. 101, 2.2.1. 130, 2.2.2. 140: If, for (X, S, JL) and (Y, T, II), / : X x Y 1--+ C is S x T-measurable then for almost every fixed x resp. y

is T-measurable resp. S-measurable. If

f

ixxY

I/(x, y)1 d(JL x II) < 00

then all the integrals in

exist and both iterated integrals are equal to

f

ixxY

/(x, y) d(JL x II).

function letter 5.1. 223 Fundamental Theorem of Calculus 2.1.2. 64 gap 2.6.2. 181: for a power series E~=o anz n , a set K ~f {k, k+ 1, ... ,k+l} such that an = 0, n E K.

Glossary/Index

275

H. G. 5.2. 230 C. F. 1.3.1. 29, 1.3.3.38,4.2.220 Gaufiian elimination 1.3.3. 38: the reduction via an invertible m x m matrix g of an m x n matrix A to an m x n matrix B ~f (bij )~;!!l ~f gA such that bij = 0 if i > j. - measure (on the algebra Z of cylinder sets in Hilbert space) 4.2. 220 Gauf3-Seidel algorithm 1.3.1. 29: a recursive method for finding approximate solutions of the matrix-vector equation Ax = b; when A is SQUARE and for an invertible P, A = P - Q and the spectral radius of p-1Q is less than 1, for an arbitrary Xo, the algorithm generates the sequence GARNIR, GAUSS,

{ xnH

~f (p-1Q)nHxO + f)p-1Q)k P-1b} k=O

, nEN

which converges to a solution of the equation. GELLES, G. 2.2.1. 110 Generalized Continuum Hypothesis 5.2. 229: For any 0, NoH = 2Na • generalized nilpotent 2.4.1. 172: in a Banach algebra, an element x such that lim

n-+oo

IIxnll: = O.

- ratio test (for an infinite series) 2.1.3. 81 - root test (for an infinite series) 2.1.3. 81 generate(s) (a u-ring) 1.1.4. 5, (a free group) 1.1.5. 11, (the commutator subgroup) 1.1.5. 15, (a group of Lie type) 1.2.2. 22, (a u-ring) 2.2.1. 105, 2.2.3. 144, (a u-ring) 4.2. 217: A set X generates an object A in a category C iff A is the intersection of the class of all C-objects containing X; the elements of X are called the generators of A. generator(s) 1.1.5. 11: see generate(s). GODEL, K. 5.1. 224, 5.1. 225 Godel's completeness theorem 5.1. 224 - undecidability (incompleteness) theorem 5.1. 225 good (topological vector space) 5.2. 229 GORENSTEIN, D. 1.1.5. 19 gradient 1.3.3. 38: in the context of linear programming, the directional derivative of the Cost function; more generally, of a function f in IRRft , the vector

of partial derivatives. J. P. 2.3.1. 162

GRAM,

Glossary/Index

276

Gram-Schmidt orthonormalization 2.3.1. 162: for a linearly independent set {Xn}nEN in a Hilbert space, the algorithm leading to the sequence def

YI

=

Xl

IIXIII def

Yn+l

Xn+1 -

= II xn+l -

L:~=l ",n

(Xn +l, (

y"J Yk )

L."k=1 Xn+b Yk Yk

graph 1.3.2. 34, 2.2.2. 142: of a function f : X 3 x the set {(x,f(x)) : XEX}.

t-+

II'

>1 n

.

f(x) E Y, in X x Y

greatest common divisor: for a pair {m, n} of natural numbers, the greatest natural number that is a factor of each; more generally, in a commutative ring R without divisors of zero, for a pair a, b of ring elements, an element c that is a factor of both a and b and such that if d is a factor of both a and b then d is also a factor of c. GROTHENDIECK, A. 2.3.1. 157 group 1.1.1. 1 - algebra 2.4.2. 175: over a locally compact group G with Haar measure 1', the set L1 (G) with multiplication defined by convolution:

group-invariant (measure) 2.2.3. 143 group of Lie type 1.2.2. 22

Haar, A. 1.1.4. 5, 2.2.1. 104 measure 1.1.4. 5, 2.2.1. 104: on a locally compact group G, a Borel measure I' that is a) translation invariant, i.e., if E is measurable and x E G then l'(xE) = I'(E), b) positive for each nonempty open set, and c) finite for every compact set. HADAMARD, J. 2.6.2. 181 Hadamard's gap theorem 2.6.2. 181: If the series S ~f L::=o anz n has a finite and positive radius of convergence R, if nl < n2 < ... ,

-

and an = 0 if n ¢ {nkhEN' then S represents a function f holomorphic in {z : Izl < R} and {z : Izl = R} is a natural boundary for f. HAHN, H. 2.3.1. 162 Hahn-Banach theorem 2.3.1. 162, 2.3.2. 162: If V is a vector space over C and p is a seminorm defined On V then a linear functional f, defined On a subspace W of V and such that If (w) I ::; p( w) on W, may

Glossary /Index

277

be extended to a linear functional F defined on V and satisfying the inequality IF(v)1 $ p(v) on V. W. 3.1.2. 198 half-space: in a vector space V and for a linear functional f : V 1-+ JR, a set of the form {x : x E V, f(x) $ r}. halting problem 5.1. 227: that of determining whether a program applied to a data-set stops (halts) after finitely many operations. HAMEL, G. 1.1.46 Hamel basis 1.1.4. 6: in a vector space V, a maximal linearly independent subset. HARDY, G. H. 2.1.1. 51 HAUSDORFF, F. 2.2.1. 112; 2.2.3. 144 Hausdorff 2.2.1. 112: of (the topology of) a space, that any two points lie in disjoint neighborhoods. - dimension 2.2.1. 122: of a subset S of a metric space (X, d) and for a positive p, HAKEN,

-

topology 2.4.1. 174: a topology in which any two points lie in disjoint neighborhoods. - -Young theorem 2.3.4. 169: If 1 1 < p $ 2, p

1

+ -p' = 1'

if f E £1' ([-11", 11"], C), and if {Cn}nEZ is the set of Fourier coefficients , 00' of f then {Cn}nEZ E lP; if {Cn}nEZ E lP then Ln=-oo enelnz normconverges in V' ([-1I",1I"],C) (to a function f in V' ([-11",11"], C)). HEDLUND, G. A. 1.1.5. 11 HELLY, E. 2.3.1. 161 Helly selection theorem 2.3.1. 161: If S ~f {Fn}nEN is a sequence of functions from [0, I] to JR and such that for some finite M, sup var(Fn) , sup IFn(O)1 nEN

< 00

nEN

then S contains a subsequence converging everywhere on [0, I] to a function F, also of bounded variation. HEMASINHA, R. 4.2. 218, 4.2. 220 HERBRAND, J. 5.1. 227 HILBERT, D. 2.1.4. 101, 2.3.1. 156, 2.3.1. 159, 3.1.1. 186

Glossary/Index

278

space 1.3.2. 36, 2.3.1. 156, 2.3.1. 159: a vector space 'H. over C and on which there is defined a positive definite, conjugate symmetric, conjugate bilinear functionalj (If 'H. is not finite-dimensional then it is assumed that 'H. is complete with respect to the metric associated to the bilinear functional.) -'s problems 2.1.4. 101 HIRZEBRUCH, F. 3.3. 208 HOLDER, 0.2.3.4. 169 Holder inequality 2.3.4. 169: For the measure situation (X, S, p.), if -

1 E V(X, C),

,

9 E V (X, C), 1 < p,

1

1

p+ pi = 1

then Ig E Ll(X,C) and IIlglil :::; II/lIpllgllp'· holomorphic 2.1.2. 61, 2.6.1. 180, 2.6.3. 182, 2.6.4. 183: of a C-valued function 1 defined in a region n of C, that f'(z} exists for all z in n. homogeneous linear differential equation 2.5.2. 177 homomorphism 1.1.3. 4: for algebraic structures A and B, a mapping h:At-+B

that, * representing the generic k-ary operations, respects them, e.g., h(a * b} = h(a} * h(b}j 1.1.4. 5: if h is between topological structures, it is assumed to be continuous. homotopic 1.3.2. 37, 3.2.2. 209: see homotopy. homotopically equivalent 3.3. 209 homotopy 1.3.2. 36: a continuous map

h: [0,1] x X :3 (t, x) whereby two maps

1:X

t-+

Y and 9 : X

t-+

ht(x}

t-+

Yare homotopic:

ho(x} == I(x}, hl(x} == g(x}. L. 2.5.2. 179 2.3.1. 158 hull 2.4.2. 175: of an ideal I in a Banach algebra, the set of regular maximal ideals containing Ij see also (closed) convex hull. hyperplane 4.3. 222: in a vector space V, a translate of the kernel of a linear functional, i.e., for a linear functional fjJ and a constant a or a vector y such that fjJ(y} = a, a set of the form HORMANDER, HUNT, R. A.

{x: fjJ(x}=a}

(= fjJ-l(O} +y).

ideal 2.4.2. 174: in a ring R, a proper subring I such that x E R => xIUIx C I.

Glossary/Index

279

identities 1.1.5. 11 identity (of a group) 1.1.1. 1 - relation 2.2.3. 147 Identity Theorem 1.3.1. 26: If f and 9 are analytic in a region n, if Sen and S has a limit point in n, and if f = 9 on S then f = 9 on n. image measure 2.2.2. 137 - - catastrophe 2.2.2. 138 implies 5.1. 223, 5.1. 224 inbreeding 2.1.2. 59, 2.2.1. 115: the process of repeating an operation, performed on the whole of a structure, on similar parts of the structure, e.g., as in Example 2.1.2.2. 58 inclusion 5.2. 229 - map 2.2.2. 138: for a subset B of a set A, the restriction to B of the identity map A 3 x H X E A, i.e., B '-+ A. indeterminates 1.3.1. 28: with respect to a ring R, symbols Xl, ... ,Xn used to form polynomials

in which the coefficients are in R. index 1.1.2. 3: for a subgroup H of a finite group G, the quotient #(G) ~fG. H

#(H)

..

- (indices) 1.3.2. 37: for a rectifiable closed curve "Y in to a point a in e \ "Y*, the integer Ind (a) .,

e and with respect

~f ~ (I d"Y(t) . 21r~

10

"Y(t) - a

When"Y is absolutely continuous Ind.,(a) is written

"Y'(t) dt ~f _1_ ( ~ 21ri 10 "Y(t) - a 21ri 1., z - a·

_1_ (I

infinitely differentiable 2.1.1. 51, 2.1.2. 61: of a function f in eRR, that f has derivatives of all orders, see differentiable. initial segment 2.2.1. 130: in a well-ordered set S, for some b in S \ inf(S) a subset of the form {c : c E S, c -< b } . injection 1.1.3. 5: a one-one map (not necessarily a surjection). injective 1.1.5. 10: of a map, that it is an injection.

Glossary/Index

280

inner measure 2.2.1. 107, 2.2.2. 138: in a measure situation (X, S, 1'), the map 1'. : 2x 3 E 1-+ sup {IL(A) : A E S, ACE} . - product 1.3.2.33, 2.1.3. 68: in a vector space V, a conjugate symmetric and conjugate bilinear functional. inscribed 2.2.1. 123: of a polyhedron II, with respect to a surface ~, that all the vertices of II are on ~. integrable 2.1.2. 64: for a measure situation (X, S, 1') and of a function f in JR. x , that f is measurable and

Ix

If(x)1 dlL <

00.

interior 2.1.1. 45. 2.3.3. 166: of a subset A of a topological space X, the union A 0 of the open subsets of A. intermediate value property 2.1.1. 43, 2.1.2. 53: of a function f in JR.[a,bl, that if a :5 c < d :5 b there is in (c, d) a e such that f(e) lies between f(c) and f(d). interpolate (a function) 2.5.2. 179: If't/J is a function on an interval [a,bj, the function F interpolates 't/J at points Xl, ••• ,Xn of [a, bj if

interval 1.1.4. 6, 2.1.1. 43 invariance of domain theorem: see Brouwer's invariance of domain theorem. inverse 1.1.1. 1, 1.3.2. 31 - of a quaternion 1.1.5. 13 - relation 2.2.3. 147 inversion formula 4.2. 219: for f in L1 (JR., C), if

f: JR. 3 t A

1

1-+ - -

h

.,fi/iJR.

e-'"t Z f(x)dx

is itself in L1 (JR., C) then, almost everywhere,

f(x) =

1 In::. V 211"

h JR.

e'"t Z f(t) dt. A

invertible 1.3.2. 31: of an endomorphism T of a vector space V that there is an endomorphism S such that ST = TS = I (~f id). is equivalent to 5.1. 224 isometric 2.2.3. 155: of a map f : (X, d) 1-+ (Y, 6) between two metric spaces, that it preserves distances: 6(f(a), f(b)) = d(a, b). isometry 1.3.2. 35: an isometric map. isomorphic 1.1.5. 10: of two algebraic structures, that there is a bijective homomorphism between them.

Glossary jIndex

281

isoperimetric inequality 2.2.1. 121: if C is a rectifiable Jordan curve in lR? and if S is the bounded component of lR 2 \ C, then A(S) :5 l(~t

.

Jacobi, C. G. J. 1.2.2.21 Jacobi identity 1.2.2. 21: in the set Matnn of n x n matrices, A 0 B denoting AB - BA, the identity: A 0 (B JAMES. R. JORDAN

0

C)

+C

0

(A

0

B)

+B

0

(C

0

A)

= O.

D. 2.3.2. 165

c. 2.2.1. 113

Jordan block 4.3. 222: a SQUARE matrix of the forIP

( ~ ~ , , :1, A

-

A

each, ;,l! QUach, ;.

o.

contour 2.6.5. 184: a Jordan curve-image. curve 2.2.1. 113: a homeomorphism 'Y : 'll'1-+ C. - -image 2.2.1. 113 "curve" theorem: Let J in lRn be the homeomorphic image of the boundary 8B of the unit ball

B 1 def = {X Then lRn \ J is the union of two disjoint regions and J is the boundary of each of them. When n = 2 the statement given is known as the Jordan curve theorem. - normal form 1.3.1. 27: for a SQUARE matrix A, a similar matrix J, i.e., for some invertible matrix P, J = p- 1 AP, consisting of Jordan blocks situated on its diagonal. - region 3.1.2. 198: a region R for which the boundary 8R is a Jordan curve-image.

Kakeya, S. 2.2.1. 124 Kakeya problem 2.2.1. 124 KARLIN, S. 2.3.1. 160 KARMARKAR, N. 1.3.3. 38 k-ary marker, representation 2.1.1. 52 kernel 1.1.3. 4: of a homomorphism if> of an algebraic structure, the inverse image of the identity, e.g., if if> is a group homomorphism, the kernel of if>

Glossary/Index

282

is q,-l(e)j if q, is an algebra homomorphism, the kernel of q, is q,-l(O)j 2.4.1. 172: of a set of regular maximal ideals in a Banach algebra, their intersection. The intersection of all regular maximal ideals in a commutative Banach algebra A is the radical of A. kernel (hull(!)) 2.4.2. 175 KLEE, V. L. 1.3.3.38 KLEENE, S. 5.1. 227 KNASTER, B. 3.1.2. 194 KOCH, J. 3.1.2. 198 KOLMOGOROV, A. N. 2.1.3.89,2.1.4. 101,4.2.216 - criteria 4.2. 216 KOWALEWSKI, S. 2.5.2. 179 KREIN, M. 2.3.2 162 Krein-Milman theorem 2.3.2. 162: A compact convex set K in a topological vector space V is the closed convex hull of the set of the extreme points ofK. KURATOWSKI, C. 3.1.2. 194

lattice 2.3.4. 169: a partially ordered set in which each pair of elements has both a least upper bound and a greatest lower bound. least upper bound 1.2.3. 23: for a subset S of an ordered set X, in X an element x such that s E B ~ s ~ x and such that if s E S ~ s ~ y then y -i. x. LEBESGUE, H. 1.1.4. 6, 2.1.2. 63, 2.1.3. 87, 2.2.1. 104 - integrable 2.1.2. 63, 2.1.3. 87: of a function f in JRRR and with respect to the measure situation (JRn, C, A), that f is measurable and that

[ If(x)1 dAn <

JRR -

00.

measurable 1.1.4. 6: of a map f in JRRR, that for every open set U in JR, f-l(U) is a Lebesgue measurable subset of JRn j 1.1.4. 6: of set Sand two Borel null sets Nl and N2 in JRn , that there is in JRn a Borel set A such that

left identity 1.1.1. 1 - inverse 1.1.1. 1 LEIBNIZ, G. W. von 2.4.1. 173 Leibniz's rule 2.4.1. 173: if D is a derivation defined on a Banach algebra then Dn(xy) =

~ (~)Dk(x)Dn-k(y).

length (of an arc) 2.2.1. 114, 2.2.1. 123

Glossary jlndex

283

length-sum 2.1.2. 56: for a set of intervals, the sum of their lengths. LEWY, H. 2.5.2. 179 LIE, S. 1.2.2. 21 Lie algebra 1.2.2. 21: a (nonassociative) algebra L over a ring R; the multiplication map L x L 3 {x, y} 1--+ [x, y] E L is such that for all x in L, [x,x] = 0 and for all triples {x,y,z} in L3 there obtains the Jacobi identity:

[x, [y, z]] + [z, [x,y]] + [y, [z,x]]

= O.

- group 2.2.1. 106: an analytic manifold that is also a topological group G in which the map G x G 3 (x, y) 1--+ xy-l has an analytic parametrization. - type 1.2.2. 21 limit point 2.1.3. 67, 2.1.3. 69, 2.6.2. 181: of a subset S of a topological space X, in X a point x such that every neighborhood of x meets S \ {x}. linear 1.2.2. 21, 1.3.2. 33: of a mapping between vector spaces, that it is a homomorphism. - function space 2.3.4. 168: a function space that is also a vector space with respect to addition of functions. - functional 2.2.1. 104: a homomorphism of a vector space into a field. - interpolation 2.1.2. 55: for a given function I in aiR, a linear function L defined on an interval [a, b] and such that L(a) = I(a) and L(b) = I(b). -(ly) isometric 2.3.2. 165: for a normed vector space, a norm-preserving homomorphism. - lattice 2.2.1. 104: a vector space that is also a lattice. - ordering 3.1.2. 192: for a set S, an ordering --< such that if x and yare in S then exactly one of x --< y, y --< x, and x = y obtains. - span 4.1. 215: of a set S of vectors, the intersection of all subspaces containing S. linearly independent 1.1.4. 6, 2.1.3. 71: of a set of vectors that no nontrivial linear combination of a (finite) subset of them is O. LIPSCHITZ, R. 2.5.2. 178 Lipschitz condition 2.5.2. 178: satisfied by a function I in a(a,b) iff for some positive a and some constant K and all x and y in (a, b), II(x) - l(y)1 :5 Klx -

ylQ·

LOBACHEVSKI, N. I. 3.1.1. 187 local extremum 2.1.4.99: for a function I : X 1--+ a defined on a topological space X, a value I(a) such that for some neighborhood N of a, x E N => II(x)1 :5 II(a)l· locally bounded 2.1.1. 47: of a function I in aX, that for each point x in X, III is bounded in some neighborhood of x. - compact 1.1.4. 5: of a topological space, that every point lies in a compact neighborhood.

Glossary/Index

284

- - group 1.1.4. 5, 2.2.1. 104: a topological group that is locally compact. - connected 3.1.2. 191: of a topological space, that it has a (neighborhood) base consisting of connected sets. - convex 5.2. 229: of a topological vector space, that it has a neighborhood base consisting of convex sets. logical connective 5.1. 223 - consistency 3.1.1. 187, 5.1. 225: of a system of axioms and rules of inference, that there is no well-formed formula A such that both A and its negation ...,A are theorems. - independence 3.1.1. 187: of a system of axioms and rules of inference, that no axiom is logically deducible from the others in the system. LORENTZ, G. G. 2.1.4 103 LOWENHEIM, L. 5.1. 224 Lowenheim's theorem 5.1. 224

Maclaurin, C. 2.1.3. 82 Maclaurin series 2.1.3. 82 MALLIAVIN, P. 2.4.2. 175 marching sequence 2.2.2. 135 MATIJASEVIC, Ju. V. 5.1. 228 matrix 1.1.5. 13: in its most primitive form, a rectangular array

of elements of a ring Rj in more general terms, for a pair r, A of sets, an element of MatrA ~f RrxA (frequently it is assumed that a matrix is a function that is zero at all but finitely many points of r x A, i.e., that a matrix is a function with compact support when r x A is viewed as a space with discrete topology)j in the language of linear algebra, if V resp. W are vector spaces over 1K and with Hamel bases

V def = { v., } .,Er resp. W def = { w~ } ~EA and if T E [V, Wj then Tv., ~f E~EA t.,~ W~j the matrix

is associated with T and the pair {V, W} of basesj Tvw is a function with compact supportj if S E [W, Uj corresponds to the matrix Swu ~f

Glossary/Index

285

(8~6) for the Hamel bases W and U~f {U6}6Ea, then the composition 80T is associated for the Hamel bases V and U with the matrix product (80 T)vu

~f

(L t-Y~8~6) ~EA

~f Tvw8wu; -yEr,6Ea

if X = Y and V = W, then the association [V] +-+ MatAA of elements of [V] with their correspondents as matrices is an anti-isomorphism of the algebra [V] of endomorphisms of V and the algebra of their associated matrices: compositions of endomorphisms are mapped into products, in reversed order, of their associated matrices. maximal biorthogonal set 2.3.1. 162 - (probabilistically) independent set 4.1. 215 maximally Q-linearly independent subset 1.1.4. 7: a set of vectors linearly independent over Q and properly contained in no Q-linearly independent set. maximum modulus theorem 2.6.4. 183: If I is holomorphic in a region of C and pEn then I/(P)I is a (local) maximum of III in a neighborhood of p iff I is constant in n. mean value theorem 2.1.4. 98: If I : an 1-+ a is differentiable then there are functions fk : an \ 01-+ a such that if h ~f (hl, .. . , hn) -:f= 0 then

n

I(x + b) - I(x) =

n

n

k=l

k=l

L Ix. (X)hk + L fk(h).

In particular, if I is differentiable in (a, b) and continuous in [a, b] there is in (a,b) a c such that I(b) - I(a) = f'(c)(b - a). measurable 1.1.4. 5, 2.2.1. 103: for a measure situation (X, S, J.L) and of a subset E of X, that E E S; 2.2.3. 144: of a group G, that there is on 2G a finitely additive measure J.L such that J.L( G) = 1 and for all 9 in G and all A in 2G , J.L(gA) = J.L(A). - group 2.2.3. 144 - map 1.1.4. 5, 2.2.2. 137: for two measure situations (X, S, J.L) and (Y, T, 11) and of a map I : X 1-+ Y, that for each E in T, 1- 1 (E) E S. measure 2.2.1. 104 - situation 2.2.1. 103: a triple consisting of a set X, a a-ring S of subsets of X, and a count ably additive map J.L : S 3 E 1-+ [0,00].

Glossary/Index

286

metric 1.3.2. 33: of a topological space X, that there is a map (a metric) d: X x X 3 (a, b)

-

1-+

[0,00)

such that a) d(a, b) = 0 <:} a = b, b) d(b, e) ::; d(a, b) + d(a, e), and c) the set of all open balls is a neighborhood base for the topology of X. density theorem 4.1. 213: If E is a Lebesgue measurable subset of]R then for almost every x in E

. A(E n (x - 6, x + 6)) _ 1 11m 21:u - .

6--+0

midpoint-convex 1.1.4. 7: of a function f

(x;

f in ]RR, that always

y) ::; f(X); f(y).

MILIN, I. M. 2.6.6. 185 Milin conjecture 2.6.6. 185 MILMAN, D. 2.3.1. 162 MILNOR, J. 1.2.2. 22, 3.3. 208 minimal base (of a topology) 2.2.1. 112 - polynomial 1.2.1. 20, 1.3.1. 26: for a SQUARE matrix A, the (unique) polynomial rnA such that a) rnA(A) = 0, b) the leading coefficient of rnA is 1, and c) the degree of rnA is the least among the degrees of all polynomials satisfying a) and b). MINKOWSKI, H. 1.3.2.33 Minkowski inequality 1.3.2. 33 MINTY, G. L. 1.3.3. 38 MITTAG-LEFFLER, G. 2.6.2. 187 Mittag-Lemer expansion 2.6.2. 187: If S has no limit points in C, there is in H (C \ S) an f having S as its set of poles and having at each point of S a prescribed principal part: If S ~f {an}l::;n::;oo and

then in a neighborhood D (an' rt of an, 00

f(z) = Pn(z)

+ L: Ck (z k=l

rn x n matrices 1.3.1. 25 model 5.1. 224 modes (of convergence) 2.2.2. 134

an)k .

Glossary /Index

287

modulo null sets 4.1. 212: of a statement that, "null sets aside," it is valid. modulus of continuity w(j, Xo, f) 2.3.1. 161 monomorphism 1.1.5. 10, 1.3.2. 31: an injective, i.e., one-one, homomorphism. monotone 2.1.1. 42 - increasing resp. decreasing 1.2.3. 24: of a map I between partially ordered sets X and Y, that Xl

:::S X2

=> I (xd

:::S

I (X2)

resp.

I (xd t I (X2) •

MOORE, E. H. 1.3.1. 28, 2.1.4. 97 Moore-Osgood theorem 2.1.4. 97 Moore-Penrose inverse 1.3.1. 28: for an m x n matrix A, an n x m matrix A+ such that AA+ A = A. MORERA, G. 2.6.1. 180 Morera's theorem 2.6.1. 180 MORSE, M. 1.1.5 11 Morse-Hedlund semigroup, 1.1.5 11 morphism: See category. MOULTON, F. R. 3.1.1.187,3.1.1.189 Moulton's plane 3.1.1. 189

n-dimensional 1.3.1. 25: of a vector space V, that it has a basis consisting of n vectors. - content 2.2.1. 104: for a set S in lRn , the Riemann integral (if it exists) of the characteristic function xs. If the integral exists S has content. - manifold 3.3. 209, 4.2. 217: a Hausdorff space X on which there is an n-dimensional differential geometric structure. natural boundary 2.6.2. 181: of a function I holomorphic in a region fl, the boundary of a (possibly larger) region in which I is holomorphic and beyond which I has no analytic continuation. neighborhood 1.1.4. 5, 1.2.3. 24: of a point P in a topological space X, a set containing an open set containing P. neighboring vertices 1.3.3. 38, 2.2.1. 123: in a polygon or a polyhedron II, vertices connected by an edge of II. net 1.2.3. 24: a function on a diset. NEUMANN, B. H. 1.1.5. 11 NEUMANN, J. VON 2.2.3. 144,5.1. 228 NEWTON, I. 1.3.1. 29, 2.1.3. 94 Newton's algorithm 1.3.1. 29, 2.1.3. 94: If I : lR t-+ lR is a differentiable function let (ao, I (ao)) be the coordinates of a point on the graph of I and assume I' (ao) ¥: 0 ¥: I (ao).

n

288

Glossary/Index

Define a sequence

{an}nEN

as follows: def

I (an)

= an - I' (an)

an+l

so long as I' (an) ¥: O. The algorithm is occasionally successful in generating a sequence {an} nEN such that del 1.Iman=a

n--+oo

exists. Furthermore, in some instances, I(a) = O. O. M. 2.2.2. 137 nilpotent 1.2.2. 21: of an element x in a ring, that for some n in N, xn = O. - semigroup 1.1.5. 12: a semigroup E containing a zero element 0, Le., for all s in E, Os = sO = 0, and such that for some k in N, every product of k elements of E is O. nonassociative algebra 1.2.2. 21 noncommutative field 1.1.5. 13 nondegenerate 2.1.1. 50: of an interval, that it is neither empty nor a single point. non-Jordan 3.1.2. 198: of a region n in a2 , that its boundary an is not a Jordan curve-image. nonmeasurable 1.1.4. 6: of a function (or a set), that it is not measurable. nonmetrizable 3.2.2. 201: of a topological space, that its topology is not derivable from (induced by) a metric. nonnegative orthant 4.3. 222: in an, the set of vectors having only nonnegative components. nonrectifiable 2.2.1. 114, 2.2.1. 123, 3.1.2. 193: of an arc ,,/, that l("() is not finite; of an arc-image "/"', that L ("(*) is not finite. norm 1.3.2. 33, 2.1.3. 68: in a vector space V, a map NIKODYM,

1111 : V

3

x 1-+ IIxll E [0,00)

such that for all x and y in V and a in C,

IIxll = 0 <=> x = 0 lIaxll = lalllxll IIx + yll :::; IIxll + IIYII; 2.4.1. 172: for a homomorphism T: A sup {IITxll

1-+

B between normed spaces,

: x E A, IIxll = 1 } .

- -induced 1.1.5. 17: of a metric d in a normed vector space V, that for all x and y in V d(x, y) ~f IIx - YII; of a topology Tin a normed vector space V, that Tis derived from the norm-induced metric.

289

Glossary /Index

- (of a quaternion) 1.1.5. 13 - -separable 2.3.2. 164: of a normed vector space V, that it is separable in its norm-induced topology. normal distribution (function) 4.2. 218: for a random variable f, the distribution function

P({w : f(w)

~

x})

=

def

1 rn= v21r

1:& -00

(t2) dt.

exp - 2

- operator (see spectral theorem): for a Hilbert space 1i, an endomorphism N such that NN* = N*N. - subgroup 1.1.2. 3: in a group G, a subgroup H such that for all x in G, xH=Hx. normalized (measure) 2.2.3. 143 normally distributed 4.2. 218: of a random variable, that its distribution function is the normal distribution. normed vector space 3.2.1. 199: a vector space endowed with a norm. NOVIKOV, P. S. 1.1.5. 11 nowhere dense 2.1.1. 43, 2.2.1. 107: of a set E in a topological space X, that X \ E = X j alternatively, that in every neighborhood of every point of X there is a nonempty open subset that does not meet E. null set 2.1.2. 56: in a measure situation (X, S, J.I.), a set of measure zero.

odd 2.1.3. 87: of a map f : V 1-+ W between vector spaces, that f( -x) - f(x)j (also, of a permutation, that it is not even). one-one (see injection): of a map f in Y x, that

=

a", b ~ f(a) '" f(b). open 1.1.4. 5, 3.2.2. 203: of a map f : X 1-+ Y between topological spaces, that the images of open sets are openj 1.1.4. 5, 2.1.1. 43: of a set U in a topological space X, that U is one of the sets defining the topology ofX. - arc-image 2.2.1. 113: in a topological space X, the image 'Y «0,1)) for a 'Y in C «0, 1), X). - ball 2.1.3. 67, 3.2.1. 200: for a point P in a metric space (X, d) and a positive r, the set {Q : Q E X, d(P, Q) < r}. optimal vertex 1.3.3. 38 - vector 1.3.3. 38 orbit 2.2.3. 147: in a set X on which a group G acts, for some P in X, a set of the form {g(P) : 9 E G}. order 1.1.2. 2: of a group G, its cardinality #(G)j 1.1.2. 4: of an element a of a group, the least natural number m such that am = ej 2.3.4. 168: in a set S, a binary relation >- such that if a, b E S then at most one of

290

Glossary/Index

a >- b, b >- a, and a = b obtains, i.e., >- is partial order; customarily >- is assumed to be transitive: a>- b/\b >- c => a >- C; if exactly one of a >- b, b>- a, and a = b obtains, >- is a total order; 2.5.1. 177: of a differential equation, the maximum of the orders of derivatives appearing in the differential equation; 3.1.1. 186: in Euclidean geometry, an axiomatized concept related to "betweenness." ordered field 1.2.3. 22 order-isomorphic 1.2.3. 23: of two ordered sets A and B, that there is an order-preserving bijection f : A 1-+ B. ordinally dense 2.2.1. 111 - similar 2.2.1. 111 orthant 1.3.3. 37 orthogonal complement 4.1. 214: of a set 8 of vectors in a Hilbert space, the set 81- of vectors orthogonal to each vector in 8. - matrix 1.2.1. 20: a SQUARE matrix over JR and in which the rows form an orthonormal set of vectors. - vectors 1.3.2. 33 orthonormal 1.3.2. 33, 2.3.1. 159: of a set of vectors in a Hilbert space, that any two are orthogonal and each is of norm one. - series 2.1.3. 84; a series E:=l an¢n in which the set {¢n}nEN is orthonormal. OSGOOD, W. F. 2.1.4. 97 outer measure 2.2.1. 107, 2.2.2. 138: for a set X, a count ably subadditive map /J* : 2x 3 E 1-+ /J*(E) E [0, co]; for a measure situation (X, S, /J), the map /J* : 2x 3 E 1-+ inf {/J(A)

A E S, E

c

A} .

Pal, J. 2.2.1. 125 Pal join 2.2.1. 125 parametric description 2.2.1. 114: for an arc 'Y : [0,1] 3 t 1-+ JRn , a continuous autojection s : [0,1]1-+ [0,1] (used to provide an arc 'f/ : [0,1] 3 t 1-+ 'Y (s(t)) such that 'f/* = 'Y*. parallelotope 2.2.1. 114 partial differential operator 2.5.2. 179: for ak1 ... k .. , kl + ... + k n ::; N, in Coo (JRn , JR), the map

°: ;

-

order 2.3.4. 168: See partially ordered.

Glossary jIndex

291

partially ordered 2.3.4. 168: of a set S, that there is defined among some or no pairs x, y in S x S an order (q.v.) denoted t and customarily subject to the condition of tmnsitivity: (x t y) 1\ (y t z) :::} x t z.

partition 2.1.2.65: of an interval I, a decomposition of I into (finitely many) pairwise disjoint subintervals; 2.5.2. 179: of a set S, a decomposition of S into a set of pairwise disjoint subsets. PEANO, G. 3.1.2. 193, 5.1. 224 PENROSE, R. 1.3.1. 28 perfect 2.1.1. 44, 2.2.1. 107, 2.2.1. 112: of a set S in a topological space, that S is closed and that every point of S is a limit point of S. period 2.1.1. 48: of a function f defined on a group G, in G an a such that for all x in G, f(xa) = f(x). periodic 2.1.1. 48: of a function f defined on a group G, that f has a period, q.v., different from the identity. permutation 2.1.3. 69: an autojection of a set. PERRON, O. 2.2.1. 124 Perron tree 2.2.1. 124 piecewise linear 2.1.2. 66, 2.1.3. 92, 2.2.2. 138, 2.3.1. 157, 2.5.2. 179: of a function f in lRlR, that there is in lR a finite sequence A ~f {an}Z'=l such that on each component of lR \ A f is a linear function. PLATEAU, J. 2.2.1. 118 Plateau problem 2.2.1. 118 POINCARE, H. 3.1.1. 187,3.3. 208 Poincare conjecture 3.3. 208 - model for plane geometry, 3.1.1. 187 point of condensation 2.1.1. 52: of a set S in a topological space X, a point P such that for every neighborhood N(P), #(N(P) n S) > #(N). polar decomposition 1.3.2. 36: for a continuous endomorphism T of a Hilbert space 1i, the factorization of T into a product of a positive definite endomorphism P and a unitary automorphism U: T = PU. polynomially dominated 1.3.3. 38: of a function f in lRIRR , that there is in lRlRR a polynomial p such that for all x in lRn , f(x) ::; p(x). positive definite 1.3.2. 33: of an inner product (x, y) that (x, x) ~ 0 and (x, x) = 0 ¢> x = 0.; 1.3.2. 36: of an endomorphism T of a Euclidean space 1i, that for all x in 1i, (Tx, x) ~ O. POST, E. 5.1. 227 power set 2.2.1. 105, 3.2.2. 200: of a set S, the set 28 of all subsets of S. predicate letter 5.1. 223 presentation 1.1.5. 11 presented 1.1.5. 11 primal linear programming problem 1.3.3. 37

292

Glossary /Index

primitive 2.1.2. 64, 2.2.2. 140: for a function f in R.IR , a function F such that F' = f. probabilistic independence 4.1. 210 probabilistic measure situation 4.1. 210 product measure 2.1.4. 101, 2.2.2. 140: for measure situations (X, S, p,) and (Y, T, v), in the measure situation (X x Y, S x T, p, xv) the a-ring S x T is generated by l' ~f {A x B : A E S, B E 7} and the measure p, x v is the unique extension to S x T of the set function

e:l' 3 A x B

1-+

J.t(A) . v(B).

- measure situation 2.2.2. 140: See product measure. - of relations 2.2.3. 147 proof 5.1. 224 proper subfield 1.1.5. 16: a field that is a proper subset of another field. - subgroup 1.1.5. 10: a subgroup that is a proper subset of another group. - inclusion 5.2. 229 pseudo-inverse 1.3.1. 28: of a matrix A, its Moore-Penrose inverse. pure quaternion 1.1.5. 13: a quaternion q ~f bi + cj + dk.

quadratic form 1.3.2. 35: on a Euclidean vector space V and for a selfadjoint endomorphism B, the function Q : V 3 x 1-+ (Bx, x) E R.. quantifier 5.1. 223 quaternion 1.1.5. 13 quotient - , e.g., quotient algebra, quotient group, quotient ring 1.1.4. 8: for a group G and a normal subgroup H, the group G I H consisting of the cosets of H and in which the binary operation is G I H x G I H 3 (xH, yH) 1-+ xyH; for a ring R or algebra A and an ideal I in R or A, the ring RI lor AI I consisting of the cosets of I and in which the binary operations are defined by those operations among the representatives in R or A. - map 1.1.5. 17: in the context of a quotient structure, say AlB, the map

A3a -

1-+

the a-coset aB of B.

norm 2.4.2. 174: for the quotient space BIM of a normed space B and a closed subspace M of B, the map

IIIIQ : BIM 3

x'

1-+

Quotient Lifting 1.1.4. 8

Rademacher, H. 4.1. 215

IIX'IlQ ~f inf {lIxll

x

E

B, xlM

= x'}.

Glossary /Index

293

Rademacher function 4.1. 215 radical 2.4.1. 172: in a commutative Banach algebra, the intersection of the set of all regular maximal ideals; alternatively, the set of all generalized nilpotent elements. - algebra 2.4.2. 176: a commutative Banach algebra in which every element is a generalized nilpotent. radius of convergence 2.1.3. 84, 2.6.2. 180: for a power series E~=o cnz n , the number R ~f limsupn ..... oo lenl--k. J. 2.2.2. 137 Radon-Nikodym theorem 2.2.2. 137 random variable 4.1. 210 range 1.3.2. 31, 2.4.1. 174: for a map T: X 1-+ Y, the set {T(x) : x EX}. rank 1.1.5. 13: the dimension of the range of a linear map T between vector spaces X and Y; of a matrix, the dimension of the span of its rows or (equivalently) the dimension of the span of its columns. rational function 1.3.2. 32: a quotient of polynomials. real-closed 1.2.2. 22: of a field K, that K is real, i.e., that there is in K no x such that x 2 + 1 = 0, and that every real algebraic extension of K is K itself. recomposition-compression 2.2.1. 126 rectifiable 2.2.1. 117, 2.6.5. 184: of an arc 'Y : [0, I] 3 t 1-+ 'Y(t) E an, that its length l('Y) is finite; of an arc-image 'Y*, that L ("(*) is finite. - Jordan contour 2.6.5. 184: a rectifiable Jordan curve in C. reduce 1.1.5. 10 reduction 1.1.5. 10 reflexive 1.1.2. 3: of a relation R, that always xRx; 2.3.1. 159: of a Banach space B, that its natural embedding in B** is surjective. - Banach space 2.3.1. 159, 2.3.2. 164 region 2.2.1. 117, 2.2.1. 131, 2.6.1. 180: a connected open subset of a topological space X. regular maximal ideal 2.1.2. 175: in an algebra A, a subset M that is a maximal ideal such that AIM has a multiplicative identity. relation(s) 1.1.5. 11, 2.2.3. 147 relatively free 2.2.1. 106: of a subset S of a group C, that if RADON,

sn,

is an abstract word and W(s,€) = e for every n-tuple s in then W(g, €) = e for every n-tuple g in In short, identities valid throughout S are those and only those valid throughout C. representation (of a number) 2.1.1. 52 resultant (of two polynomials) 1.3.1. 27 retrobasis 2.3.1. 159: in the dual space B* of a Banach space B, a Schauder basis {bn}nEN for which the set {b;}nEN of associated coefficient functionals lies in B regarded as a subspace of B**.

cn.

Glossary /Index

294

B. 2.1.1. 51, 2.1.1. 54, 2.1.1. 64, 2.1.1. 69, 2.1.3. 85, 2.2.1. 104, 2.2.3. 145 Riemann derangement theorem 2.1.3.69 - integrable 2.1.2. 54, 2.1.2. 64, 2.2.1. 104: of a function f defined on a product I of intervals in JR n , that RIEMANN,

1

f(Xb .•.

,xn ) dx

exists, i.e., that f is bounded and that Discont(f) is a null set. - sphere 2.2.3. 145 Riemann-Lebesgue theorem 2.1.3. 85: If f E L1 ([-11",11"], C) and inx /71" f(x) e-In::. dx, n E Z -71" V 211" = OJ 2.1.3. 88: If f E L1 (JR, C) and d f

~

Cn

then limlnl_oo Cn

•

f(t)

1 "t = . In::.intlRf(x)e-' x dx

def

V

211"

then limltl_oo j(t) = o. (A natural generalization of the RiemannLebesgue theorem is valid for a locally compact abelian group G endowed with Haar measure J.I. defined on the (T-ring S(K) generated by the set K of compact subsets of G: If'll.' is regarded as an abelian group with respect to multiplication of complex numbers, if

G ~f {a

: a a homomorphism of G into 'lI.', }

and if f E L1 (G, C) then the (Gelfand-) Fourier transform

j : G3 a

1-+

fa f(x)a(x) dJ.l.(x) ~f j(a)

vanishes at infinity, i.e., if f > 0 there is in G a compact set K(f) such that Ij(a)1 < f if a ¢ K.) RIESZ, F. 2.3.4. 169 Riesz-Fischer theorem 2.3.4. 169: If (X, S, J.I.) is a measure situation and {4>~} ~EA is a complete orthonormal set in L2 (X, C) then L~EA c~4>~ converges in L2 (X, C) iff L~EA Ic~ 12 < 00. RIESZ, M. 2.3.4. 169 - -Thorin theorem 2.3.4. 169: Assume

a, {3, (Ti, Pj > 0, 1 ~ i def A def = ()m,n aij i,j=1' x =

=

(

m, 1

sup ( ",n

#0

L..Jj=1

~

X1,···,X n

(L~1 (TiIXil~

def

MOt ,13

~

t

Ot· pjlxjl l.) a

j ~n )

,

X

= (Xb

def

... ,

X) m

= AX t

def

Glossary jlndex

295

Then on every line in ~ {(a,.8) 0 < a ~ 1,0 <.8 ~ a} M a ,f3 is a multiplicatively convex function of its variables, i.e., if

(ab .8d, (a2' .82) E ~, t E (0,1), and

(a,.8)

= t(ab .81) + (1 -

t)(a2' .82)

then If one carefully interprets the formulae in the limiting case in which a = .8 = 0 the result remains valid. right identity 1.1.1. 2 - inverse 1.1.1. 2 rigid motion 2.2.3. 144: in a metric space (X, d) a local isometry, i.e., for a subset S of X, a map I : S -+ X such that a, bE S ~ d (a, b)

= d (f(a), I(b)).

ROBERTSON, M. S. 2.6.6. 185 Robertson conjecture 2.6.6. 185 ROBINSON, R. M. 2.2.3. 146 ROSSER, J. B. 5.1. 225 ROUCHE, E. 2.6.5. 184 Rouche's theorem 2.6.5. 184: Let "( be a rectifiable Jordan curve such that "(. and the bounded component C of C \ "(. lie in a region O. If I, U E H(O) and III > lui on "(. then I and I + U have the same number of zeros in C. row vector 1.3.1. 26: a 1 x n matrix (as opposed to a column vector, which is an n x 1 matrix). RUBEL, L. A. 2.5.2. 178 rule of inference 5.1. 224

satisfiability 5.1. 224 satisfiable 5.1. 224 - in some model 5.1. 224 SCHAUDER, J. 2.3.1. 156 Schauder basis 2.3.1. 156 SCHMIDT, E. 2.3.1. 162 SCHWARZ, H. A. 1.3.2. 33, 2.2.1. 123 Schwarz inequality 1.3.2. 33 Scottish book 2.3.1. 157 second category 2.1.1. 43, 2.1.1. 51: in a topological space a set that is not of the first category.

296

Glossary /Index

section 2.2.1. 110: in a Cartesian product X~EAX~, for AO in A, X~~~oX~. segment 2.3.3. 167: in a vector space V, the convex hull of two vectors. self-adjoint 1.3.2. 35: of an endomorphism T of a Hilbert space 'H, that for all x and y in 'H, (Tx,y) = (x,Ty); of a matrix (aij)~j~l that aij = aji. semicontinuous (lower resp. upper) 2.1.1. 48, 2.2.2. 138, 2.3.4. 171: of a function I defined in the neighborhood of a point a in a topological space X, that

I(a)

= liminf I(x) resp. z--+o

I(a)

= limsupl(x); z--+o

of a function I defined throughout X, that I is lower resp. upper semicontinuous at each point of X; equivalently that for every t in JR,

1-1 [(t, 00)] resp. 1-1 [( -00, t)] is open. Occasionally the qualifiers lower, upper are omitted and semicontinuous is used alone. semigroup 1.1.5. 11: a set S on which there is defined a binary operation subject to the sole requirement of associativity. seminorm 2.4.1. 174, 5.2. 229: for a vector space V, a subadditive map p: V 3 x

1-+

[0,00).

semisimple 2.4.1. 172: of a commutative Banach algebra B, that its radical is {OJ. separable 2.2.1. 117, 3.2.2. 201: of a topological space X, that it has a countable base of neighborhoods; 2.1.3. 66: of a metric space (X, d), (equivalently) that it contains a countable dense subset. separate, separating 2.1.4. 102, 5.2. 229: A set S of functions in yX separates or is separating if whenever Xl :f:. X2 E X there is in S an I such that I (xI) :f:. I (X2). separated locally convex vector space 5.2. 229: a locally convex Hausdorff vector space. sequence (of groups) 1.1.3. 4 set 5.2. 229 sfield 1.1.5. 13, 1.2.1. 19 shift operator 1.3.2. 32 short exact sequence of groups 1.1.4. 8 SIERPINSKI, W. 2.1.3. 70, 2.2.1. 104, 2.2.1. 130,2.2.3. 146 u-finite(ness) 2.2.2. 137, 2.3.1. 162: of a measure situation (X, S, 1'), that every element of S is the countable union of sets of finite measure. - -ring 1.1.4. 5, 2.2.1. 103: a set S of sets closed with respect to the formation of set differences and countable unions.

Glossary /Index

297

significant domain 2.1.2. 58: for a function I in aX, the set X \ 1-1 (0). signum function (sgn) 2.1.2. 64 simple 1.1.6. 18: of a group G, that only {e} and G are normal subgroupsj 1.3.1. 26: of a zero a of a polynomial p that

p(z)

= (z -

a)

II (z -

ri)j

ri#a

2.2.1. 104: of a function, that it is a linear combination of characteristic functions of measurable setsj 2.2.1. 113: of an are, that its defining map '"Y is injective. - arc-image 2.2.1. 113 - closed curve 2.2.1. 113 - open arc-image 2.2.1. 113 simplex algorithm (in linear programming) 1.3.3. 38 simplification 1.1.5. 10 simplify 1.1.5. 10 simply connected 2.6.3. 183, 3.2.2. 205: of a set S in a topological space X, that every closed arc '"Y : [0,1]1-+ S is homotopic in S to point, i.e., to a constant function. simultaneously diagonable 1.3.1. 27: of a set S of n x n matrices, that there is a fixed invertible matrix T such that for each A in S, T-l AT is a diagonal matrix. singular 1.3.1. 25: of an endomorphism T of a vector space, that T is not invertible. singularity 2.6.2. 181: of a function I holomorphic in a region n, in the closure of n a point a such that there is no definition of I(a) that allows I to be holomorphic at a. skew field 1.1.5. 13, 1.2.1. 19 SMALE, S. 1.3.3.38,3.3. 209 smooth 2.1.2. 63: of a function, that it is infinitely differentiable. SOLOVAY, R. 5.2. 229 solvable 1.1.4. 8: of a group G, that it contains a sequence {Gk}k=O such that: a) Go = G and Gn = {e}j b) Gk is a normal subgroup of Gk-b 1 ~ k ~ nj c) Gk - 1 /G k is abelian, 1 ~ k ~ n. somewhere differentiable 2.3.3. 166 source (in the theory of free topological groups) 1.1.5. 17 span 1.1.5. 16, 2.3.3.168,4.1. 214: of a set S of vectors, the set of all (finite) linear combinations of vectors in Sj (a set S in a vector space V spans a subspace W if W is the span of S: W = span(S)). spectral measure 1.3.2. 36: with respect to a Hilbert space 1t and a measure situation (X, S, It) in which XES, a map E : S 1-+ 'P into the set 'P of self-adjoint projections in [1t]j it is assumed that: a) E is count ably

298

Glossary /Index additive; b) if

Ai E S, i = 1,2 Mi ~f E(Ai)'H., i

= 1,2

Mv ~f span (Ml U M 2 ) M" ~f span (Ml n M 2 ) and if Ev resp. E" is the orthogonal projection on Mv resp. M" then

-

c) E(0) = 0 and E(X) = I. radius 1.3.1. 29: of an element (an n x n matrix M) of [an], the number PM equal to the maximum of the set of absolute values of the eigenvalues of M; 2.4.1. 172: of an element x of a Banach algebra B, sup {IAI : A E O'(x)} = lim IIxnll~. n ..... oo

-

(If [an] is regarded as a Banach algebra the two definitions just given are equivalent.) theorem 1.3.2. 36: If N is a normal operator in ['H.], if P is the set of self-adjoint projections in ['H.], and if B2 is the set of Borel sets in C, there exists a spectral measure E : B2 ~ P such that N

=

fc

zdE(z).

spectrum 1.3.2. 31: for an element x in a Banach algebra B with identity e, the set O'(x) of complex numbers A such that x - Ae is not invertible. splits 1.1.3. 5: said of an exact sequence

if Hand G x K are isomorphic. 101

SPRECHER, D. A. 2.1.4. STEINITZ, E. 2.1.3. 70

stereographic projection 2.2.3. 145: the map

a2

(

3 x, Y

)

~

(2 (x2x y x2 + y2 - .25) S + y2 + .25)' 2 (x2 + y2 + .25)' 2 (x2 + y2 + .25) E !

STIRLING, J. 2.1.1. 52, 2.5.1. 176 Stirling's formula 2.1.1. 52, 2.5.1. 176:

1·

n!

n~~ (~t J21rn

1

= .

Glossary/Index

299

stochastic process 4.2 216: a set {f~} of random variables defined with respect to a probabilistic measure situation (X, S, P). STOKES, G. G. 2.2.1. 121 Stokes's theorem 2.2.1. 121 STONE, M. H. 2.1.3. 94, 2.1.4. 102,2.2.2. 133,2.3.3. 165,2.6.4. 183 Stone-WeierstraB theorem 2.1.3. 94, 2.1.4 102, 2.2.2. 133, 2.3.3. 165, 2.6.4. 183: If X is a compact Hausdorff space and if A is a separating algebra of continuous lR-valued functions on X then A is norm-dense in C(X,lR). STRAUS, E. G. 1.1.2. 3 strictly increasing 2.1.2.56: of a function f in lRR, that a < b =* f(a) < f(b). subadditive: of a function f from an abelian semigroup to an ordered abelian semigroup, that f(x + y) ~ f(x) + f(y). subgroup 1.1.2. 3: in a group G, a subset H that is, with respect to the group operation in G, a group. subspace 2.1.3. 70: (of a vector space V), in V a subset W that is also a vector space with respect to the operations in V; (of a topological space X), in X a subset Y that is topologized by heredity, e.g., a set G in Y is open (in Y) iff G is the intersection of Y and an open subset ofX. sum, i.e., to sum via a summability method 2.1.3. 74 summability method 2.1.3. 74 sum set 2.2.1. 109: for two subsets A and B of a group resp. an abelian group the set

{ab : a E A, b E B} resp. {a + b : a E A, b E B } . superbifurcation 2.5.2. 179: the failure of a differential equation to have at most a finite number of solutions at some point. support: of a function f defined on a topological space X and taking values in a field K, the closure of the set where f is not zero: supp(f) ~f {x : f(x) -IO}. supremum 1.1.5.14: ofaset {O~hEA of topologies for a set X, the topology U~EA O~; 1.2.3. 23: see least upper bound. surjection 1.1.3. 5: a map T : X 1-+ Y such that T(X) = Y. SWITCH 1.3.3. 39: in the simplex algorithm applied to linear programming, an exchange: basic variable +-+ free variable. SYLVESTER, J. J. 1.3.2. 35 Sylvester's Law of Inertia 1.3.2. 35 symmetric 1.1.2. 3: of a relation R, that xRy =* yRx. - difference 4.1. 212: for two sets A and B, the set A6B ~f (A \ B) l:J (B \ A).

Glossary/Index

300

- group 1.1.2. 3, 1.1.4. 8: for n in N, the group Sn of permutations of the set {1,2, ... ,n}.

tamely embedded sphere 3.2.2. 205 tame sphere 3.2.2. 205 TARSKI, A. 1.3.2. 22, 2.2.3. 145, 5.1. 224 ternary marker 2.1.1. 52 the abelianization of a group G 1.1.5. 15: the quotient group of G by its commutator subgroup Q(G). - free group 1.1.5. 10 - k-ary representation (of a number) 2.1.1. 52: for a number x in JR, if the k-ary representation of x is unique, the k-ary representation of Xj

if there are two k-ary representations, the one in which, for some N in N, all k-ary markers Xn are k - 1 if n ~ N. theorem 5.1. 224 thick 2.2.1. 107, 4.2. 218: of a set E in a measure situation (X, S, J.L), that the outer measure of X \ E is zero. THOMPSON, J. 1.1.6. 18 THORIN, G. O. 2.3.4. 169 TOEPLITZ, 0.2.1.3.75 Toeplitz matrix 2.1.3. 75 TONELLI, L. 2.2.2. 140 Tonelli's theorem 2.2.2. 141 topological division algebra 1.2.3. 25: a topological space A that is a division algebra such that A x A 3 ( a, b)

1-+

a - bE A

and A x (A \ {O}) 3 (a,b)

1-+

ab- 1 E A

are continuous. - group 1.1.4. 5: a topological space G that is a group such that G x G 3 (x, y) 1-+ xy-l EGis continuous. - vector space 2.3.4. 168: a vector space V for which the additive structure is a topological group and that is defined over a topological field K so that the map K x V 3 (a, x) 1-+ ax E V is continuous. topology 2.1.1. 42: for a set X, in the power set 2x a subset 0 containing 0 and X and closed with respect to the formation of finite intersections and arbitrary unions of the elements of O.

301

Glossary/Index

topopathology 3.2.2. 200: a topological phenomenon that goes counter to the expectations of many mathematicians; the study of such phenomena. totally disconnected 2.2.1. 112, 3.1.2. 193: of a topological space, that it contains no connected subsets other than the empty set and individual points. - a-finite: of a measure situation (X, S, J.I.), that X is the countable union of sets of finite measure (whence XES); by abuse of language the adjective a-finite is used to describe both X and J.I. when (X, S, J.I.) is a-finite. total variation 2.1.2.65: for a function I in lR.[a.b1, n

sup a=XO<Xl < .. ·<xn=b

L

II (Xj) - I (Xj-I) I, n

E

N.

j=1

trajectory 2.2.3. 147: in a set X on which a group G acts, for some P in X, a set of the form {g(P) : 9 E G}; see orbit. transcendental 2.2.1. 108: of a number z in C, that it is not the zero of a polynomial having coefficients in Z. transition matrix 4.3. 221 transitive 1.1.2. 3: of a relation R, that xRy 1\ yRz => xRz.

translate 2.2.1. 125: of a subset E of a space acted upon by a group G and for an element x of G, the subset xE def = { xy: y E E } . translation invariant 2.2.1. 104: of a measure situation (G, S, J.I.) for a group G, that for all x in G and all E in S, J.I. (xE) = J.I.(E). transpose 1.3.1. 25: for a matrix (aij )7,';;:;1' the matrix (ajir;,ir;:,I' trigonometric polynomial 2.1.3. 89: a function p of the form n

p: lR. 3 x

L ak cos kx + bk sin kx, ak, bk

1--+

E

C, n E N.

k=O

- series 2.1.3. 84: a series of the form 00

L ak cos kx + bk sin kx, ak, bk E C. n=O

trivial (instances of probabilistic independence) 4.1. 211: (for independence of sets), for any event A the triple {0,X,A}; (for independence ofrandom variables), for any random variable I and any constant function c the pair {c, - topology 3.2.2. 200: for a space X, the topology 0 consisting of exactly and X.

n.

o

302

Glossary/Index

true 5.1. 224 truth 5.1. 224 T-sum 2.1.3. 75 TUKEY, J. 2.3.3. 167 TURING, A. 5.1. 227 Thring machine 5.1. 227 two-sided sequences of complex numbers 1.3.2. 32

ultimately constant 3.2.2. 201: of a net {X~hEA' that for some Ao in A, there is an a such that x~ = a if A >- Ao. unbounded 1.1.4. 6: of a function f in aX, that either

= xEX inf f( x) (f is unbounded below),

a) -

00

b)

= sup f(x) (f is unbounded above),

00

xEX

or both a) and b) obtain. unconditional basis 2.3.1. 159: in a Banach space E, a Schauder basis

such that for each x in E and every permutation'll" of N, 00

L b;(n) (x)b,..(n) n=l

convergesj equivalently, for every sequence

{En

En

= ±1, n EN},

00

LEnb;(x)bn n=l

converges. uncountable 1.1.4. 6, 2.1.1. 52: of a set S, that #(S) > # (N). undecidable 5.1. 225 uniform modulus of continuity 2.3.1. 161 - structure 1.2.3. 24: for a set X, in 2xxX a subset U such that: i. ii. iii. iv.

U, V E U :::} 3W {W E U, We Un V}j U E U :::} U J {(x, x) : x EX} ~ 6j UEU:::}U-l~f{(y,x): (X,y)EU}EUj WoW denoting {(x, z) : 3y {(x, y), (y, z) E W}}, {U E U} :::} 3W {{W E U} 1\ {W 0 W c Un.

uniformity 2.1.4. 97: a uniform structure, q.v.

Glossary/Index

303

uniformly bounded variation 2.3.1. 160: of a set {f~hEA of functions, that there is an M such that for all A, Tf). ~ M. - continuous 1.1.4. 6: of a map f : X 1-+ Y and for uniform structures U for X and V for Y, that if V E V there is in U a U such that (a, b) E U ~ (f(a), f(b)) E V. unit ball 2.3.4. 171: in a metric space (X, d), a set ofthe form {x : d(x, a) ~ I}.

unitary 1.3.2. 36: of an automorphism U of a Euclidean space 'H, that for all x and y in 'H, (Ux, Uy) = (x, y). univalent 2.6.6. 184: of a holomorphic function, that it is injective. universal comparison test 2.1.3. 72 vanishes at infinity 2.1.3. 88, 2.3.2. 163: of a C-valued function f defined on a topological space X, that for every positive e there is in X a compact set K(e) such that x ¢ K(e) ~ If(x)1 < e. variable 5.1. 223 variation 2.3.1. 160: See total variation. - lattice: See linear lattice. - space 1.1.4. 6, 1.3.1. 25: an abelian group V that is a module over a field II{, i.e., there is a map II{

xV

3

(a, x)

1-+

a· x E V

such that a) a· (b· x) = ab· x and b) a· (x + y) = a· x elements of V are vectors. VESLEY, R. 5.1. 228 vicinity 1.2.3. 24: an element U of a uniform structure U.

+ a· y.

The

Walsh, J. L. 4.1. 215 Walsh function 4.1. 215: for a set {rk 1 , ••• , rknJ of Rademacher functions, the function m

IIrk;. i=l

°

weak 2.5.2. 179: of a solution of a differential equation, that it is a distribution; 3.2.2. 202: of a topology for a space X and a set {f~hEA of is generated by the maps from X into a topological space Y, that set {f;l(V) : V open in Y}. . weaker 3.2.2. 201: of a topology 0, that it is a subset of another topology

°

0'. weakest 2.3.2. 162: of a topology 0, that it is weaker than each topology of a set of topologies.

Glossary/Index

304

WEIERSTRASS, K. 2.2.2. 139, 2.6.4. 183 WeierstraB approximation theorem 2.2.2. 139, 2.6.4. 183: If K is a compact subset of an, E > 0, and I E C(K,a) then there is a polynomial p: an 1-+ a such that on K, II - pi < E. - infinite product representation 2.6.2. 181: If S1 is a region in C, if

A ~f {an}nEN

C S1 \

{O},

if A has no limit points in S1, and if A is the set Z f of zeros of

I

in

H(S1), each zero listed as often as its multiplicity, then in N there is a sequence {mn}~=l' in H(S1) there is a function g, and in NU{O} there is a k such that for z in S1,

I(z) = zk exp (g(z))

II (1- ~) exp (~+- ... + (~)mn) . an an an

nE N

M-test 2.1.3. 83: If E:=llanl < 00 and if I/n(x)1 :5 lanl, n E N,x E X then E:=ll/n(x)1 converges uniformly on X. weight (of a topological space) 2.2.1. 112 well-formed formula 5.1. 223-4 well-ordered 2.2.1. 130: of a totally ordered set S, that in every nonempty subset T of S there is a least element t, i.e.,

-

(T

=f: 0) 1\ (x E T)

:::} (t = x) V (t

-< x).

WEYL, H. 1.3.2. 36 Weyl minmax theorem 1.3.2. 36: If A is a self-adjoint n x n matrix and if its eigenvalues are Al :5 A2 :5 ... :5 An then

Aj

=.

min max (Ax, x). dlm(v)=n-(j-l) xEv,lIxll=l

wildly embedded sphere 3.2.2. 206 wild sphere 3.2.2. 206 word 1.1.5. 9 - problem 1.1.5. 11 WRONSKI, H. 2.5.1. 177 Wronskian 2.5.1. 177

x-left coset 1.1.2. 3 x-right coset 1.1.2. 3

Young, W. H. 2.3.4.169

Glossary/Index

305

Zermelo, E. 5.2. 228 Zermelo-Fraenkel 5.2. 228: of the set of axioms provided by Zermelo and Fraenkel as the foundation for set theory. zero homomorphism 2.4.2. 175: the homomorphism mapping each element of an algebra into O. ZORN, M. 1.1.4. 6 Zorn's lemma 1.1.4. 6: If (8,~) is a partially ordered set in which each linearly ordered subset has an upper bound, then 8 has a maximal element, i.e., there is in 8 an s such that for any s' in 8, either s' ~ s or s' and s are not comparable, i.e., never s ~ s'.