Fbgbkl_jkl\hh[s_]hbijhn_kkbhgZevgh]h h[jZah\ZgbyJhkkbckdhcN_^_jZpbb
...
206 downloads
203 Views
103KB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
Fbgbkl_jkl\hh[s_]hbijhn_kkbhgZevgh]h h[jZah\ZgbyJhkkbckdhcN_^_jZpbb
HIHGYLBYOLHIHEH=BQ?KDH=H IJHKLJ:GKL<:BG?IJ?JU
KhklZ\bl_ev X?=ebdebo
2
<?GB? Ex[hc q_eh\_d bamqZ\rbc gZqZeZ fZl_fZlbq_kdh]h ZgZebaZ ihgbfZ_l \Z`ghklv ihgylby g_ij_ju\ghklb nmgdpbb G_fgh]h mi jhsZy kblmZpbx fh`gh kdZaZlv qlh g_ij_ju\ghklv nmgdpbb - wlh fZl_fZlbq_kdZy nhjfZebaZpby ke_^mxs_]h k\hckl\Z ]jZnbd wlhc nmgdpbbfh`ghgZjbkh\ZlvgZebkl_[mfZ]bg_hlju\ZydZjZg^ZrZ l_]jZnbdgb]^_g_ jZaju\Z_lkyNmgdpby_klvqZklguckemqZc[h e__h[s_]hihgylbyhlh[jZ`_gbydhlhjh_hij_^_ey_lkym`_g_^ey qbk_e Z^eywe_f_glh\ijhba\hevguofgh`_kl\<:JBL?EVGU?KHH;J:@?GBY GZihfgbf deZkkbq_kdh_ hij_^_e_gb_ g_ij_ju\ghklb qbkeh \hcnmgdpbbf\lhqd_x\hkoh^ys__dDhrb
3
Hij_^_e_gb_ Nmgdpby fgZau\Z_lkyg_ij_ju\ghc\lhqd_ x_keb^eyex[h]hε > kms_kl\m_lδ=δ(ε) >lZdh_qlhbalh ]hqlh| x - x’| < δ , ke_^m_lqlh| f(x) - f(x’)| < ε. <\_^_ggh_\ur_hij_^_e_gb_^himkdZ_lfh^bnbdZpbxm^h[ gmx^ey^Zevg_cr_]hbaeh`_gby Hij_^_e_gb_’. NmgdpbyfgZau\Z_lkyg_ij_ju\ghc\lhqd_ x_keb^eyex[hchdj_klghklbUlhqdbf(x)kms_kl\m_lhdj_kl ghklvVlhqdbxlZdZyqlhbalh]hqlhlhqdZx’ijbgZ^e_`bl Vke_^m_lqlhf(x’) ijbgZ^e_`blU. G_ljm^gh\b^_lvqlh^eyqbkeh\uonmgdpbcHij_^_e_gb_b Hij_^_e_gb_’wd\b\Ze_glguihkdhevdmkh^ghcklhjhgufgh`_ kl\hlhq_d{x’} lZdboqlh| x - x’| < δy\ey_lkyhdj_klghklvxlhqdb xgZau\Z_fhcδ-hdj_klghklvxx khhl\_lkl\_gghfgh`_kl\hlhq_d {y} lZdboqlh| f(x) - y| < εy\ey_lkyhdj_klghklvxlhqdbf(x), gZ au\Z_fhcε-hdj_klghklvxf(x) Zk^jm]hcklhjhgu\gmljbex[hc hdj_klghklb U lhqdb f(x) kh^_j`blky ε-hdj_klghklv ^ey ^hklZ lhqgh fZeh]h ε khhl\_lkl\_ggh \ ex[hc hdj_klghklb V lhqdb x kh^_j`blkyδ-hdj_klghklv^ey^hklZlhqghfZeh]hδ). JZkkfhljbf^\Zfgh`_kl\ZXb Y=h\hjylqlhaZ^Zghhlh [jZ`_gb_F: X → Y_kebaZ^ZghijZ\beh aZdhg ihdhlhjhfmdZ` ^hfmwe_f_glmxba XihklZ\e_g\khhl\_lkl\b_we_f_gly = F(x) ba Y Qbkeh\Zy nmgdpby y\ey_lky gZb[he__ ba\_klguf ijbf_jhf hlh[jZ`_gby<wlhfkemqZ_h[uqghX = Y = R -fgh`_kl\h\_s_kl \_gguoqbk_e qbkeh\ZyijyfZy ZaZdhgFaZ^Z_lkynhjfmehc:gZ ijbf_j \_s_kl\_gghfm qbkem x klZ\blky \ khhl\_lkl\b_ \_s_kl \_ggh_qbkeh sin x \wlhfkemqZ_F _klvnmgdpbykbgmk
4
Ihgylb_hlh[jZ`_gbyhij_^_e_gh^eyex[hciZjuijhba\hev guofgh`_kl\H^gZdhfh`gheb\ijhba\hevghfkemqZ_^Zlvhij_ ^_e_gb_g_ij_ju\ghklbFihZgZeh]bbkHij_^_e_gb_fbebHi j_^_e_gb_f’?G_ljm^gh\b^_lvqlhwlh]hk^_eZlvg_evayihkdhev dm gZ ijhba\hevguo fgh`_kl\Zo g_l gb ihgylby hdj_klghklb bk ihevam_fh]h \ Hij_^_e_gbb ’ gb ihgylby δ-hdj_klghklb εhdj_klghklb bkihevam_fh]h\Hij_^_e_gbbLZdqlh^ey\\_^_gby dhjj_dlgh]h hij_^_e_gby ihgylby g_ij_ju\ghklb F fu ^he`gu eb[h\\_klbij_^\Zjbl_evghihgylb_hdj_klghklb\hh[s_eb[hih gylb_ ε-hdj_klghklb GZ ijbf_j_ qbkeh\uo nmgdpbc \b^gh qlh εhdj_klghklby\eyxlkyqZklgufkemqZ_fhdj_klghkl_c\hh[s_b_k ebfuohlbf^ZlvgZb[he__h[s__hij_^_e_gb_g_ij_ju\ghklbfu ^he`gu khkj_^hlhqblv k\h_ \gbfZgb_ gZ dhjj_dlghf \\_^_gbb ih gylbyijhklhhdj_klghklblhqdb\ijhba\hevghffgh`_kl\_ Fgh`_kl\hgZdhlhjhf“ijZ\bevgh” \\_^_ghihgylb_hdj_kl ghklb gZau\Z_lky lhiheh]bq_kdbf ijhkljZgkl\hf Ih^q_jdg_f lj_[h\Zgb_qlh[ufgh`_kl\h[uehlhiheh]bq_kdbfijhkljZgkl\hf y\ey_lkyfbgbfZevguf^eylh]hqlh[u[uehdhjj_dlghhij_^_e_gh ihgylb_g_ij_ju\gh]hhlh[jZ`_gby 1. HIJ?>?E?GB?LHIHEH=BQ?KDH=HIJHKLJ:GKL<:B I?J
5
bgl_j\Zekp_gljhf\wlhclhqd_ l_ε-hdj_klghklv^ey^hklZlhqgh fZeh]h ε p_ebdhf \oh^bl \ wlh fgh`_kl\h >ey hldjuluo fgh `_kl\\uihegyxlky^\Z\Z`guok\hckl\Z h[t_^bg_gb_ex[h]h ^Z `_[_kdhg_qgh]h gZ[hjZhldjuluofgh`_kl\_klvhldjulh_fgh`_ kl\h b i_j_k_q_gb_ dhg_qgh]h qbkeZ hldjuluo fgh`_kl\ _klv hl djulh_fgh`_kl\h HdZau\Z_lky_kebg_dhlhjucgZ[hjfgh`_kl\h[eZ^Z_lwlbfb k\hckl\Zfblhkfgh`_kl\ZfbbamdZaZggh]hgZ[hjZfh`ghjZ[hlZlv \hfgh]hflZd`_dZdkh[uqgufbhldjulufbfgh`_kl\Zfb>Z^bf lhqgu_hij_^_e_gby JZkkfhljbfijhba\hevgh_fgh`_kl\h; Hij_^_e_gb_GZ[hj τih^fgh`_kl\fgh`_kl\ZXgZau \Z_lkylhiheh]b_c_kebhgh[eZ^Z_lke_^mxsbfbk\hckl\Zfb L ;bimklh_fgh`_kl\h\oh^yl\ τ LL h[t_^bg_gb_ex[h]hqbkeZ fgh`_kl\ ba τ ijbgZ^e_`bl τ LLL i_j_k_q_gb_ dhg_qgh]h qbkeZ fgh`_kl\baτijbgZ^e_`blτ?kebgZ[hjτaZ^Zg;gZau\Z_lky lhiheh]bq_kdbfijhkljZgkl\hfZ\oh^ysb_\τfgh`_kl\ZgZau \Zxlkyhldjulufb Ijbf_jhf lhiheh]bq_kdh]h ijhkljZgkl\Z y\ey_lky qbkeh\Zy ijyfZy k fgh`_kl\Zfb hldjulufb \ h[uqghf kfuke_ >_ckl\b l_evgh \ky qbkeh\Zy ijyfZy hq_\b^guf h[jZahf hldjulZ imklh_ fgh`_kl\h\dexqZxl\qbkehhldjuluoihhij_^_e_gbx wlhg_ijh lb\hj_qb\h - ihkdhevdm \ imklhf fgh`_kl\_ g_l lhq_d lh fh`gh kqblZlvqlhdZ`^Zybagbo \oh^bl\imklh_fgh`_kl\hkg_dhlh jhc ε-hdj_klghklvx DZd m`_ kdZaZgh \ur_ LL b LLL \uiheg_gu Lhiheh]bx khklhysmx ba h[uqguo hldjuluo fgh`_kl\ gZ qbkeh \hcijyfhc[m^_fgZau\Zlvh[uqghclhiheh]b_c
6
Ijb\_^_f_s_^\Zijbf_jZGZex[hf;jZkkfhljbflhiheh ]bx\dhlhjhc\k_]h^\Zfgh`_kl\Z \k_;bimklh_LZdZylhiheh ]bygZau\Z_lkyljb\bZevghcIjhlb\hiheh`gZykblmZpby-gZex[hf ;\dexqbf\lhiheh]bx\hh[s_\k_ih^fgh`_kl\Z; \qZklghklb \k__]hlhqdb kZfh;bimklh_ih^fgh`_kl\hWlZlhiheh]bygZ au\Z_lky^bkdj_lghc H[jZlbl_ \gbfZgb_ qlh ljb\bZevgmx b ^bkdj_lgmx lhiheh ]bx fu aZ^Zeb hibkZ\ \k_ \oh^ysb_ \ gbo fgh`_kl\Z K h[uqghc lhiheh]b_cfug_kfh]ebwlhk^_eZlvbgZfijbrehkvhibku\Zlv__ kihfhsvxk\hckl\Zdhlhjhfmm^h\e_l\hjyxl__fgh`_kl\ZQlh[u ba[_`Zlvwlh]hg_m^h[kl\Z[ueh\\_^_ghihgylb_[Zaulhiheh]bb Hij_^_e_gb_GZ[hjhldjuluofgh`_kl\SgZau\Z_lky [Zahc lhiheh]bb τ _keb ex[h_ fgh`_kl\h ba τ _klv \hafh`gh [_kdhg_qgh_ h[t_^bg_gb_fgh`_kl\baS. ;Zahch[uqghclhiheh]bbgZijyfhcy\eyxlkyε-hdj_klghklb >_ckl\bl_evgh h[uqgh_ hldjulh_ fgh`_kl\h oZjZdl_jbam_lky l_f qlh dZ`^Zy _]h lhqdZ bf__l g_dhlhjmx ε-hdj_klghklv \oh^ysmx \ wlhfgh`_kl\hLZdqlhhq_\b^ghqlhkZfhfgh`_kl\h_klvh[t_^b g_gb_mdZaZgguoε-hdj_klghkl_c\k_o_]hlhq_d Ijb\_^_f_s_^\Zijbf_jZI_j\ucbagbo-lhiheh]byAZjbk kdh]h gZ qbkeh\hc ijyfhc - bgl_j_k_g djhf_ \k_]h ijhq_]h l_f qlh\hagbd\j_ZevghcfZl_fZlbq_kdhcaZ^Zq_Zg_dZdwdahlbq_kdbc ijbf_j^eymq_[gbdZ<wlmlhiheh]bx\dexq_gu\kyijyfZybimk lh_ fgh`_kl\h Z lZd`_ \k_ fgh`_kl\Z gZ ijyfhc ^hiheg_gby ^h dhlhjuokhklhylbadhg_qgh]hqbkeZlhq_d Ke_^mxsZy lhiheh]by gZ qbkeh\hc ijyfhc khklhbl ba \k_c ijyfhcbimklh]hfgh`_kl\ZZlZd`_\k_ohldjuluobgl_j\Zeh\\b ^Z a, +∞ ]^_a -lhqdZijyfhc WlZlhiheh]bygZau\Z_lkyijZ\hc
7
Hlf_lbfqlh\lhqghklbZgZeh]bqghfh`ghaZ^Zlvbe_\mxlhiheh ]bx Lhiheh]byfh`_lgZke_^h\ZlvkyGZijbf_j\iehkdhklbbf_ _lkylhiheh]bykhklhysZybah[uqguohldjuluofgh`_kl\ ZgZeh ]bqgh kemqZx qbkeh\hc ijyfhc Lh]^Z gZ e_`Zs_c \ iehkdhklb ijyfhc \hagbdZ_l lhiheh]by \ dhlhjhc hldjulufb fgh`_kl\Zfb y\eyxlkyi_j_k_q_gbykwlhcijyfhcfgh`_kl\hldjuluo\iehkdh klb WlZ lhiheh]by gZau\Z_lky bg^mpbjh\Zgghc < jZkkfZljb\Z_ fhfijbf_j_bg^mpbjh\ZggZylhiheh]by-wlhh[uqgZylhiheh]bygZ ijyfhc <g_dhlhjuokemqZyojZaebqgu_lhiheh]bbgZh^ghfblhf`_ fgh`_kl\_fh`ghkjZ\gb\Zlvf_`^mkh[hc=h\hjylqlhlhiheh]by τgZXkbevg__lhiheh]bbσgZlhf`_fgh`_kl\__keb\k_fgh `_kl\Z\oh^ysb_\σ\oh^yllZd`_b\τHq_\b^ghqlhex[Zylh iheh]by kbevg__ q_f ljb\bZevgZy Z ^bkdj_lgZy - kbevg__ ex[hc lhiheh]bbLZd`_ihgylghqlhh[uqgZylhiheh]bykbevg__q_flh iheh]byAZjbkkdh]hbq_fijZ\Zylhiheh]byb\lh`_\j_fylhiheh ]bxAZjbkkdh]hbijZ\mxlhiheh]bxkjZ\gblvf_`^mkh[hcg_evaygb h^gZ ba gbo g_ y\ey_lky [he__ kbevghc q_f ^jm]Zy [he__ lh]h ^hdZ`bl_ qlh _keb g_dhlhjh_ fgh`_kl\h qbkeh\hc ijyfhc \oh^bl kjZam \ h[_ wlb lhiheh]bb lh wlh eb[h \ky qbkeh\Zy ijyfZy eb[h imklh_fgh`_kl\h Hij_^_e_gb_ Hdj_klghklvxlhqdb\lhiheh]bq_kdhfijh kljZgkl\_ gZau\Z_lky ex[h_ hldjulh_ fgh`_kl\h kh^_j`Zs__ mdZaZggmxlhqdm Hq_\b^gh qlh \ h[uqghc lhiheh]bb ihgylb_ hdj_klghklb m^h\e_l\hjy_l^Zgghfmhij_^_e_gbx
8
Bkihevamy \\_^_ggh_ hij_^_e_gb_ hdj_klghklb g_ljm^gh ^h dZaZlv ke_^mxs__ k\hckl\h hldjuluo fgh`_kl\ ex[h]h lhiheh]b q_kdh]hijhkljZgkl\Z fgh`_kl\hAhldjulhlh]^Zblhevdhlh]^Z dh]^ZdZ`^ZylhqdZxbaAbf__lhdj_klghklvp_ebdhf\oh^ysmx \ A >hdZ`bl_ wlh ml\_j`^_gb_ kZfhklhyl_evgh H[jZlbl_ \gbfZ gb_qlhoZjZdl_jbklbq_kdh_k\hckl\hh[uqguohldjuluofgh`_kl\ gZqbkeh\hcijyfhcy\ey_lkyqZklgufkemqZ_fwlh]hml\_j`^_gby G?IJ?JU:QBLHIHEH=BB ImklvaZ^Zghhlh[jZ`_gb_F: X → Y]^_XbY -lhiheh]bq_ kdb_ijhkljZgkl\Zklhiheh]byfbkhhl\_lkl\_gghτbσIhkdhev dmfu\\_ebhij_^_e_gb_hdj_klghklblhqdb\lhiheh]bq_kdhfijh kljZgkl\_ fu fh`_f ^Zlv hij_^_e_gb_ g_ij_ju\ghklb F \ lhqd_ iheghklvxZgZeh]bqghHij_^_e_gbx’. Hij_^_e_gb_Hlh[jZ`_gb_FgZau\Z_lkyg_ij_ju\guf\ lhqd_x∈X_keb^eyex[hchdj_klghklbU∈σlhqdbf(x)\Ykm s_kl\m_l hdj_klghklv V∈τ lhqdb x \ X lZdZy qlh ba lh]h qlhlhqdZx’ijbgZ^e_`blVke_^m_lqlhf(x’) ijbgZ^e_`blU. Hij_^_e_gb_ 6.Hlh[jZ`_gb_g_ij_ju\gh_\dZ`^hclhqd_ x fgh`_kl\Z XgZau\Z_lkyg_ij_ju\gufgZX. < kemqZ_ dh]^Z fgh`_kl\h X aZnbdkbjh\Zgh fu [m^_f gZ au\Zlvhlh[jZ`_gbyijhklhg_ij_ju\gufbg_mdZau\ZyX. G_ij_ju\gu_ hlh[jZ`_gby oZjZdl_jbamxlky ke_^mxsbf k\hckl\hf L_hj_fZ 1. Hlh[jZ`_gb_F: X → Yg_ij_ju\ghlh]^Zblhev dh lh]^Z dh]^Z ^ey ex[h]h hldjulh]h fgh`_kl\Z U∈σ ijh kljZgkl\ZY_]hijhh[jZaV=F-1(U)ijbgZ^e_`bl τl_y\ey _lkyhldjuluffgh`_kl\hflhiheh]bq_kdh]hijhkljZgkl\ZX.
9
>hdZaZl_evkl\h Imklv F g_ij_ju\gh l_ m^h\e_l\hjy_l Hij_^_e_gbx
F-1(U)lhih
Hij_^_e_gbx dZ`^h_ x bf__l hdj_klghklv Vx lZdmx qlh F(Vx)⊂U Ba ihke_^g_]h \dexq_gby \ qZklghklb ke_^m_l qlh Vx⊂VlZddZdihhij_^_e_gbxV_klvfgh`_kl\h\k_olhq_dxba XlZdboqlhF(x)∈ULh]^ZV=∪x∈VVx>_ckl\bl_evghlZddZddZ `^h_xijbgZ^e_`blk\h_fmVx, ∪x∈VVx kh^_j`bl\k_x,l_\dex qZ_l\k_[yVK^jm]hcklhjhgulZddZd\k_Vx kh^_j`Zlky\ V, lh b bo h[t_^bg_gb_ kh^_j`blky \ V. Ba ^\mo \dexq_gbc V ⊂ ∪x∈VVx bV ⊃ ∪x∈VVx ke_^m_ljZ\_gkl\hV=∪x∈VVxLZdbfh[jZahf V _klv h[t_^bg_gb_ hldjuluo fgh`_kl\ Vx lh _klv hgh kZfh hl djulhihk\hckl\m(ii) lhiheh]bb L_i_jvimklv^eyex[h]hhldjulh]hfgh`_kl\ZUlhiheh]b q_kdh]hijhkljZgkl\ZY l_U∈σ fgh`_kl\hV=F-1(U) hldjulh\ X l_ijbgZ^e_`blτ IhdZ`_fqlh\uiheg_ghHij_^_e_gb_\ dZ`^hclhqd_x∈X
10
L_hj_fZlZd`_iha\hey_lkljhblvgh\u_lhiheh]bbke_^mx sbf h[jZahf Imklv aZ^Zg g_dhlhjuc deZkk hlh[jZ`_gbc {F} ba fgh`_kl\ZX\qbkeh\mxijyfmxRkh[uqghclhiheh]b_c beb\ ex[h_ ^jm]h_ lhiheh]bq_kdh_ ijhkljZgkl\h - \ wlhf kemqZ_ dhgkl jmdpbyZgZeh]bqgZ AZ^Z^bfgZ[hjτih^fgh`_kl\\X\dexqb\ lm^Zfgh`_kl\Z\b^ZF-1(U)^ey\k_ohldjuluofgh`_kl\U\Rb ^ey \k_o hlh[jZ`_gbc F ba {F} \k_ bo h[t_^bg_gby b dhg_qgu_ i_j_k_q_gbyZlZd`_\k_Xbimklh_fgh`_kl\hIhemq_ggucgZ [hjτ [m^_llhiheh]b_cIjbwlhfihL_hj_f_baihkljh_gbyke_ ^m_lqlh\k_hlh[jZ`_gbyba{F}[m^mlg_ij_ju\gufbIh^h[gu_ lhiheh]bbqZklhbkihevamxlkybhdZau\Zxlky\_kvfZihe_agufb Hij_^_e_gb_ 7. Hlh[jZ`_gb_ F ba lhiheh]bq_kdh]h ijh kljZgkl\ZX\lhiheh]bq_kdh_ijhkljZgkl\hYgZau\Z_lky]h f_hfhjnbafhf _keb \uiheg_gu ke_^mxsb_ ljb mkeh\by i) F g_ ij_ju\gh ii) F\aZbfghh^ghagZqgh l_^eyex[h]hy∈Ykms_kl \m_lx∈XlZdh_qlhF(x)=ybmdZaZggh_x_^bgkl\_ggh \qZkl ghklbkms_kl\m_lh[jZlgh_hlh[jZ`_gb_F-1: Y → X ); (iii hlh [jZ`_gb_F-1g_ij_ju\gh ?kebkms_kl\m_l]hf_hfhjnbafF: X →Y, lh]h\hjylqlhX b Y]hf_hfhjngu^jm]^jm]m<wlhfkemqZ_fufh`_fgZeh`blv X gZ Y [_akZfhi_j_k_q_gbcbjZaju\h\ijbde_b\Zyx∈X d F(x)∈Y. LZdqlhihemqZ_lkyqlhXbY mkljh_guh^bgZdh\h Ihgylby ]hf_hfhjnbafZ b ]hf_hfhjnghklb y\eyxlky p_g ljZevgufb ^ey fgh]bo jZa^_eh\ lhiheh]bb \ dhlhjuo bamqZxlky oZjZdl_jbklbdb hibku\Zxsb_ ]hf_hfhjngu_ ijhkljZgkl\Z Ih kdhevdm ]hf_hfhjngu_ ijhkljZgkl\Z mkljh_gu h^bgZdh\h kf \u r_ lh bo fh`gh g_ jZaebqZlv l_kqblZlvjZagufbwda_fieyjZfb h^gh]hblh]h`_h[t_dlZKms_kl\m_ldjueZlZynjZaZqlhlhiheh]
11
fZl_fZlbd aZgbfZxsbcky lhiheh]b_c - wlh q_eh\_d g_ hlebqZx sbc [m[ebd hl qZcghc qZrdb aZ^ZqZ ihkljhcl_ ]hf_hfhjnbaf f_`^m[m[ebdhfbqZrdhckh^ghcjmqdhc WlhhagZqZ_lqlhgZb [he__h[sb_ lhiheh]bq_kdb_ k\hckl\Z[m[ebdZbqZrdbh^bgZdh\u hgbl_e_kgubbf_xlh^gm^ujdm >jm]b_ jZa^_eu lhiheh]bb bamqZxl oZjZdl_jbklbdb g_ij_ ju\guo hlh[jZ`_gbc b g_dhlhju_ ^jm]b_ Ijb wlhf qZklh ihemqZ xlky j_amevlZlu \Z`gu_ ^ey ijbeh`_gbc GZijbf_j m^Z_lky \u qbkeblv g_dhlhju_ oZjZdl_jbklbdb g_ij_ju\guo hlh[jZ`_gbc \oh^ysbo \ hij_^_e_ggu_ mjZ\g_gby dhlhju_ ihdZau\Zxl bf__l eb wlh mjZ\g_gb_ j_r_gb_ Wlh hq_gv \Z`gh \ kemqZyo dh]^Z y\gh j_rblv mjZ\g_gb_ g_\hafh`gh g_ m^Z_lky gZclb nhjfmem ^ey j_ r_gby D:DB?;U<:XLLHIHEH=BB BlZd \ ijhba\hevghf lhiheh]bq_kdhf ijhkljZgkl\_ fu fh `_f \hij_^_e_gguoij_^_eZo jZ[hlZlvlZd`_mki_rghdZdgZqb keh\hc ijyfhc b wlbf lhiheh]bq_kdb_ ijhkljZgkl\Z ihoh`b ^jm] gZ ^jm]Z H^gZdh dZ`^h_ lhiheh]bq_kdh_ ijhkljZgkl\h h[eZ^Z_l ki_pbnbq_kdbfb k\hckl\Zfb dhlhju_ bgh]^Z j_adh hlebqZxlky hl k\hckl\qbkeh\hcijyfhc Ba\_klgh lZd gZau\Z_fuo Zdkbhf hl^_ebfhklb ba dhlhjuo fu ijb\_^_f ljb ijhkl_crbo Hlf_lbf qlh qbkeh\Zy ijyfZy k h[uqghclhiheh]b_cm^h\e_l\hjy_l\k_fZdkbhfZfIjhkljZgkl\Z m^h\e_l\hjyxsb_lhevdhg_dhlhjufbagbo_kl_kl\_gghhlebqZxl kyhlg__k\hbfbk\hckl\ZfbBlZd :dkbhfZL0. >eyex[uo^\mog_kh\iZ^Zxsbolhq_dohly[u h^gZbagbobf__lhdj_klghklvg_kh^_j`Zsmx^jm]mxlhqdm
12
Hq_\b^ghqlh^eyljb\bZevghclhiheh]bb:dkbhfZL0g_\u ihegy_lky \ wlhc lhiheh]bb _klv jh\gh h^gh g_imklh_ hldjulh_ fgh`_kl\h-\k_;ihwlhfm\k_;[m^_l_^bgkl\_gghc\hafh`ghc hdj_klghklvx ^eyex[hclhqdbb^eyijhba\hevghciZjulhq_dbo ³ex[u_´hdj_klghklbijhklhkh\iZ^Zxley ex[uo ^\mo g_ kh\iZ^Zxsbo lhq_d dZ`^Zy bagbobf__lhdj_klghklvg_kh^_j`Zsmx^jm]mxlhqdm G_ljm^gh \b^_lv qlh ijhkljZgkl\h m^h\e_l\hjyxs__ L1, m^h\e_l\hjy_lbL0Zg_m^h\e_l\hjyxs__L0g_m^h\e_l\hjy_lbL1. LZd qlh ijhkljZgkl\h k ljb\bZevghc lhiheh]b_c g_ m^h\e_l\hjy_l L1Qbkeh\ZyijyfZykijZ\hclhiheh]b_clh`_g_m^h\e_l\hjy_lL1. >_ckl\bl_evghimklvx < yLh]^Z\ay\x < a < yfuihemqbfqlh (a, ∞) kh^_j`bly l_y\ey_lky_]hhdj_klghklvx bg_kh^_j`blx hlkx^Zke_^m_l\uiheg_gb_L0 H^gZdh^eyex[h]hb < xbgl_j\Ze (b, ∞) kh^_j`blb xb yl_ex[Zyhdj_klghklvlhqdbxkh^_j`bl by. Hlf_lbfqlhqbkeh\ZyijyfZyklhiheh]b_cAZjbkkdh]hm^h\ e_l\hjy_lL1>_ckl\bl_evgh^eyx ≠ yhdj_klghklvxlhqdbxg_ kh^_j`Zs_cyy\ey_lky^hiheg_gb_R\y, Zhdj_klghklvxlhqdbyg_ kh^_j`Zs_cxy\ey_lkyR\xE_]dh\b^_lvqlhijyfZykh[uqghcb ^bkdj_lghclhiheh]byfbm^h\e_l\hjyxlL1. :dkbhfZL2. >eyex[uo^\mog_kh\iZ^Zxsbolhq_dmdZ`^hc bagbofh`gh\u[jZlvihhdj_klghklblZdqlh[uwlbhdj_klgh klbg_i_j_k_dZebkv Ihgylghqlhba\uiheg_gbyL2ke_^m_l\uiheg_gb_L1Z_keb g_\uihegy_lkyL1lhg_\uihegy_lkybL2.
13
Qbkeh\Zy ijyfZy k lhiheh]b_c AZjbkkdh]h g_ m^h\e_l\hjy_l L2>_ckl\bl_evghihkdhevdm\wlhclhiheh]bbhldjulh_fgh`_kl\h hij_^_ey_lky dZd fgh`_kl\h ^hiheg_gb_ ^h dhlhjh]h khklhbl ba dhg_qgh]h qbkeZ lhq_d Z \ijyfhcqbkehlhq_d[_kdhg_qghlhex [u_^\Zhldjuluofgh`_kl\Z \lhfqbke_ex[u_^\_hdj_klghklb i_j_k_dZxlkyih[_kdhg_qghfmqbkemlhq_d Hq_\b^gh ihdZ`bl_ qlh ijyfZy k h[uqghc b k ^bkdj_lghc lhiheh]byfbm^h\e_l\hjyxlL2. Bf_xlky_s_ZdkbhfuL3bL4gZdhlhjuofu\wlhcklZlv_g_ hklZgZ\eb\Z_fky <ebygb_ Zdkbhf hl^_ebfhklb gZ k\hckl\Z lhiheh]bq_kdbo ijhkljZgkl\fuijhbeexkljbjm_fgZijbf_j_ihgylbyij_^_eZih ke_^h\Zl_evghklbbamqZ_fh]h\klZjrbodeZkkZordheu<lhiheh ]bq_kdhf ijhkljZgkl\_ hij_^_e_gb_ ij_^_eZ \u]ey^bl ke_^mxsbf h[jZahf kjZ\gbl_kh[uqgufhij_^_e_gb_f Hij_^_e_gb_ LhqdZ x ∈ X gZau\Z_lky ij_^_ehf ihke_^h \Zl_evghklb{x1, x2,…, xn,…},_keb^eyex[hchdj_klghklbU lhqdb xkms_kl\m_lghf_jN=N(U)lZdhcqlh^ey\k_on > N lhqdbxn e_`Zl\U. GZijbf_j \ h[uqghc lhiheh]bb gZ ijyfhc ij_^_ehf ihke_ ^h\Zl_evghklb {1,
1 1 1 , ,…, ,…} y\ey_lky lhqdZ ^ey ³ihklhyg 2 3 n
ghc´ ihke_^h\Zl_evghklb{a, a,…, a,…} (a -nbdkbjh\Zggh_qbkeh ij_^_ejZ\_ga,bihke_^h\Zl_evghklv{1, 2, 3,…, n,…} gZlmjZev gucjy^ g_bf__lij_^_eZ<h[uqghclhiheh]bbij_^_eihke_^h\Z l_evghklb fh`_l [ulv lhevdh h^bg _keb hg \hh[s_ kms_kl\m_l b hggZoh^blkydZd[u³jy^hf´klhqdZfbihke_^h\Zl_evghklb
14
>ey ijhkljZgkl\ g_ m^h\e_l\hjyxsbo dZdbf-gb[m^v Zdkbh fZfhl^_ebfhklbk\hckl\Zij_^_eh\fh]ml[ulv\_kvfZg_h[uqgu fb Ml\_j`^_gb_ <ijZ\hclhiheh]bbgZijyfhcex[ZylhqdZ b < ay\ey_lkyij_^_ehfihke_^h\Zl_evghklb{a, a,…, a,…}. >_ckl\bl_evghhdj_klghklvlhqdbb\ijZ\hclhiheh]bb_klv fgh`_kl\h\b^Z(c, ∞)]^_c < bIhkdhevdmb < a, (c, ∞)kh^_j`bl alh_klv\k_qe_guihke_^h\Zl_evghklb{a, a,…, a,…}LZdbfh[jZ ahfb - ij_^_e L_i_jv jZkkfhljbf ijyfmx k lhiheh]b_c AZjbkkdh]h A^_kv bf__lky_s_[he__\i_qZleyxsbcijbf_jij_^_eZihke_^h\Zl_evgh klb Ml\_j`^_gb_<lhiheh]bbAZjbkkdh]hex[ZylhqdZx ∈ R y\ey_lkyij_^_ehfgZlmjZevgh]hjy^Z >_ckl\bl_evgh aZnbdkbjm_f ijhba\hevgmx hdj_klghklv U lhqdbxIhhij_^_e_gbxlhiheh]bbAZjbkkdh]h^hiheg_gb_U^h R khklhblbadhg_qgh]hqbkeZlhq_dIhkdhevdm\gZlmjZevghfjy^m [_kdhg_qgh_qbkehlhq_dhlkx^Zke_^m_lqlh\Ukh^_j`blky[_k dhg_qgh_ qbkeh _]h lhq_d lh _klv gZqbgZy k g_dhlhjh]h N \k_ lhqdbn > N e_`Zl\U . H[uqgZyb^bkdj_lgZylhiheh]bbm^h\e_l\hjyxlZdkbhfZfL0 -L2b\gbog_kms_kl\m_lklhevwdahlbq_kdboijbf_jh\ij_^_eh\ H^gZdhg_ke_^m_l^mfZlvqlh^bkdj_lgZylhiheh]byhq_gvihoh`Z gZh[uqgmxGZihfgbfqlh\^bkdj_lghclhiheh]bbhldjulufy\ ey_lkyex[h_fgh`_kl\hl_\qZklghklbex[ZylhqdZy\ey_lkykZ fZk\h_chdj_klghklvxIhgylghqlh\wlhfkemqZ_\hdj_klghklbx lhqdb x g_l lhq_d hlebqguo hl x lh _klv ex[Zy nbdkbjh\ZggZy
15
lhqdZxfh`_l[ulvij_^_ehflhevdhlZdboihke_^h\Zl_evghkl_cm dhlhjuogZqbgZykg_dhlhjh]hN\k_qe_guxn>N jZ\gux. Bf__lky_s_h^gZ\Z`gZykbkl_fZZdkbhfhlghkbl_evghdhlh juodklZlbjZaebqZxlky^\_ihke_^gb_lhiheh]bb Hij_^_e_gb_ =h\hjylqlhlhiheh]bq_kdh_ijhkljZgkl\h m^h\e_l\hjy_l \lhjhc Zdkbhf_ kq_lghklb _keb lhiheh]by wlh]h ijhkljZgkl\Z bf__l [Zam khklhysmx ba kq_lgh]h qbkeZ fgh `_kl\ l_ fgh`_kl\Z \oh^ysb_ \ wlm [Zam fh`gh aZgmf_jh \ZlvgZlmjZevgufbqbkeZfb H[uqgZy lhiheh]by gZ ijyfhc bf__l kq_lgmx [Zam - wlh εhdj_klghklbkjZpbhgZevgufεp_gljZfbdhlhjuoy\eyxlkyjZpbh gZevgu_lhqdb dZdba\_klghfgh`_kl\hjZpbhgZevguoqbk_ekq_l gh >bkdj_lgZylhiheh]bygZijyfhcg_bf__lkq_lghc[Zau \ex [mx [Zam wlhc lhiheh]bb ^he`gu \oh^blv \k_ lhqdb ijyfhc Z dZd ba\_klgh wlh fgh`_kl\h [he__ q_f kq_lgh _]h g_evay aZgmf_jh \Zlv
16
EBL?J:LMJ: 1. <\_^_gb_\lhiheh]bx/;hjbkh\bqX=;ebagydh\GFBajZb e_\bqY:b^jFhkd\Z GZmdZNbafZlebl -k 2. Klbgjh^ G Qbgg M I_j\u_ ihgylby lhiheh]bb Fhkd\Z Fbj 1967.-k 3. NjZgkbk >` Dgb`dZ k dZjlbgdZfb ih lhiheh]bb Fhkd\Z Fbj 1991.-k 4. Nmdk>;Nhf_gdh:L=ml_gfZo_j<E=hfhlhibq_kdZylhih eh]byFhkd\Z Ba^Zl_evkl\hF=M -k
KhklZ\bl_ev=ebdeboXjbc?\]_gv_\bq J_^Zdlhj;mgbgZ?>
