1. 1. . . . . . 1997 /
2. 2. This work is subject copyright. rights are reserved, whether the who1e or part of the materia1 is concerned, specifically, those of trans1ation, reprinting, broadcasting, reproduction by photocopying machine or simi1ar means, and storage data banks. by S. Negrepontis, Th. Zachariades, . Ka1amidas, V. Farmaki, 1988. or - : . - . & .. . 80 . . 77.10.548 - 77.02.033. FAX: 77.10.581
3. 3. ,. J.L. KeZley 1955 : [ ] : . , ::, KCf:l . : , , , . . : Bire ( ) Tychonoff . . , , , Bire Tychonoff, ' : Hhn-Bgnch ( ). Stone- Weierstrss. , , : = + Hhn-Bnch.
4. 4. , , . , .. re-1 , 4, , 15, 17, 18. . , , . ( 650 ), . . . , , , , , . ' , ' . ' , ( Grthendeck ), 17, . , , , . , . . , ' , - , , q., . , 30 1988 .., .., .., ..
5. 5. 1. 1 8 22 26 32 2. 41 41 Banach 50 Cantor, Baire, 52 59 ' 64 67 3. 71 72 78 Hahn-Banach 92 Baire: , 99 Banach Schauder 104 Hilbert 125 132 4. Baire-l 143 144 Baire- 148 157
6. 6. 5. 159 160 165 Cantor 168 171 175 6. 180 , , , 181 , , , 193 205 209 7. 216 230 8. 23~ 235 242 250 9. 256 270 10. - - 273 274 279 284 295 303 11. 311 328 12. 333 335 , 339 Lindelof 348 c.c.c. 350 354 359 13. 362 376
7. 7. 14. 379 379 Stone- Weierstrass 385 Ascoli 389 390 397 ' 40 404 409 15. 419 Eberlein Namioka 422 Amir-Lindenstrauss, Gulko 426 16. Tychonoff 436 Tychonoff 437 Stone-Cech 439 A~coli 11 443 445 17. Tychonoff - 450 450 457 462 Banach Schauder 11 473 482 18. Namioka - 489 Szlenk 489 494 Radon-Nikodym 497 : , 500 504 507
8. 8. 1. , , ( , , , , ) , , . . , , . t : Ix-yl~lx-zl+lz-yl x,Y,zEIR. , , . , , , , () , . , .
9. 9. 2 1.1. . ' . : XX-1R : () (x,y)~O ,, () (,) = = , () (,)=(,) , ( ), () (,) ~ (,) +(,) ,, ( ). ( ()) (,) ( ) . (,) . ' 2 ,..., (, ), (, 2 ),... ;. , , , ( , , ). , . , . 1.2. . ( IR). : IR IR - IR, (,) = - . , , , , IR. , , . 1.3. . ( IRk k = 1,2,...). () k = ,2,... , 2, : IR IR - IR, k (,) = - , 2(,)= (i=t xi -Yi 12)1/2 1=1 , (,) = max { - : = ,2,...,k}, =(XI ,...,Xk), =( "",Yk) IRk. , IRk 2 (), () () ) 1.1 . 2
10. 10. 3 Cauchy-Schwarzo 'ooo,Xk, ,000,Yk [ : k k k IR f(z) = 2 + 2 + =1 =l = f(z) ~ IRo ' , 2(,) ~ 2(,) +2(,) = ( ,000,Xk), = ( ,0~0'Yk), z=(zJ,ooo,Zk) ElRk, Cauchy-Schwarz J- J'000' Xk - Zk, - J,000, Zk - Yk 2 , (lRk ,2) k- lO () ~ . ( ) . ~
11. 16. 9 Sp(x,E) , , S(x, ). S(x,E)={yEx: (,) S(x, ) = ( -, + ). , (, ), , IR < , - + 2 - - - 2 () k- IRk, = ( '''',Xk) IRk > k S(X,E) = fy = ( '''',Yk): ( _)2 < 2 }. =1 k =2 S(X,E) (. ) k = 3 (. ).
12. 17. () (lRk, ,) ( 1.3), = (, '''',Xk) > k S(,) = { =(, '''''Yk): Xi -Yi S(,) = {} = ~ 1, 1< . , , , , . ' , ) . 1.12. . (,) C . ( ( ) .
13. 20. 13 (1.10) (1.12) OK~ : 1.13. . (, ) , C ;.: > S(,)=S(,)n. 1.14. . (, ) , > . S(,), > , S(y, ) C S(,). . S(,). EIR, O , S(,)CG. G=UGS(, ). ( S (, ) ct .
14. 22. 15 , , , 1.21. . (, ) , C , G C . G G = n . . ( ~ ) ' G , G, > , S (, ) C G. ' G = UYE G S(,). , 1.13, G = (U YEG Sp (,)) n , , = UyEG S(,), . ( ~ ) G. , G C , > Sp (, ) C . , 1.13, S(, ) = Sp (, ) n C n =G. ' G . , , ' ' ' . , ' , . ' > , , . , 1.17, (' ) : , 1.22. . (, ) C . ( ) . intpA ( ) intxA ( ). = U { C : G G C }. C . (1.22) 1.17 > : 1.23. . (,) C . : (i) . (ii) = .
15. 23. 16 1.24. . (, ) C ={: > S(,)C}. . > S(x, ) C , , S(,)C . . . G G C , G , > S(, ) C G C . 1.25. . () , , [,J, (,J, [,) (, ), , IR, (, ). () , , J S(, ) S(,)CS(,I)C. () k- IRk, {X=(XI,...,Xk)ElRk: l~x+"'+x~ }. . ' S(, ) n # 0 > . ~ . F , C F F. ' -- F, - F , > S(X, ) C -- F. S(,)C--, S(,)n=0, . . . > S(,)n=0. C ---S (, ). F = --- S(x, ). F AC F. , (1.3 ), F, . 1.34. . () ) IR, , (, ), (, ], [,) [,], ,IR , ( - , + ) n = 0. () IR, , ( - , + ) n ) S(,)nS(0.,1)#0, X=(XI,,,,,Xk)ElRk, X+"'+X~> , =x+"'+X~-I>O S(,)nS(0,1)=0. () k- IRk, k = 1,2,.", { = ( '''',Xk) Rk: ~ + '" + x~ < 2} {x=(xI,,,,,Xk)ElRk: l~xT+"+x~~2}.
16. 27. 20 ( ) ( ) . 1.35. . (, ) , 0 =1= C . inf{p(x,y):YEA}. (,). 1.36. . (, ) , 0 =1= C , (,) = . . ( ~) (,)=>. , ' (,) ~ , S (, ~) n = 0, ( 1.33). ( ~ ) > (,) . S(,). xfAO, --. --. , x~AO, S(,) n (--) =1= 0 > . ' (--)- ( 1.33). ( --) - = -- '. () (--). > S(,) C --. S(x,t) n = 0, ~ , --. --. ~ . > , S(, ) n = 0, S(, ) C -- . (--). (--)=--. , ' . , . 1.38. . (,) C . (i) ' , . , > (S(,),,{})n==0. () ' . > S(,)n={}. '.
17. 28. 21 1.39. . () ' , , , 1.11 (). () IR , [,), (,], [,] (, ), , IR < , [, ] . () IR : ' = ' = 0 IR' = {(-,+),,-{})I={(-,+),,-{}) g-=0, x~lN' '. IR, , (--,+),,{} 1R#0, (-,+),{}nQ#0, (-, +) ,{x}I(1R ,,(1))#0. 1.40. . (, ) , C . , > , S(, ) n . . > (S(,)-{})n={l,...,n }. = min {(, ),..., (,n )} > . (S(,)--{})n = 0, . 1.41. . (, ) C = ' U . . 1.33 1.38, ' UACA. ' . ~ , .33, (S(,)-{})#0 >, '. ACA' UA. (,) C . C C . , . 1.42. . (, ) D C . D D = . . . 1.43. . (,) D C . D D , ( D n S(x, ) # 0 > ) D n G # 0 G . . 1.19, 1.33 (1.42). 1.44. . . 1.45. . () IR ,
18. 29. 22 , ~ IR ( 1.34 ()). () IRk k = ],2,.... , ~k, IRk , IRk . () ~ < +00 ,P ( 1.5) . = { = (x)'P: Xn . 00 > Y=(Yn)E..e p . l, IYnI P < 2' n=N+1 n=I,2,,,.,N qnE~, Iqn-YnI P , ( , ) < ~ (xn ) limxn = Xn - . n , (xn ) , , (( ,)) .. 1.47. . () , J , . () ' (xn ) k- IRk , k = ,2,..., Xn = ( ,...,X~) = 1,2,..., = ( ,...,xk ) IRk . lim Xn = lim X~ = , n n = ,:..,k. lim = , - ~ 2 (Xn ,) = ,2,...,k, n J, , limxA = n =1 ,2,...,k. lim X~ = = 1,2,...,k, > n . . () X~ - 1 . - < 2' . ' =+ 2 limx~ = = 1,2,... ,, m(n)E m (x~,x) 2 < 2 ' m~m(n). mo=max{m(I),...,m(N)}. m~mo limxm = . m (:: , ) + 2 =+ (x~,x) < ~ + 2 2 2' =. ' , , . ( ) () 1.1 : (, ) = = . , ' () . () 1.1. 1.49. ( ). (, ) ( ) . - - =. ' ' '...... ' (,) > ' , . -r- = 2 . ,2 I, (,)< ~I (,)< ~2' = max { , 2}' ~ (, ) ~ (, ) + (,) < 2 = (,) . 1.49. . .
19. 32. 25 ( , ) . , Xn =, = ,2,..., . Xn = (_I)n, = ,2,..., . 1.49 : ' , . , ' (, ), 0# C (,) = ( 1.36). , . 1.50. . (, ) , 0# C . (xn) lim Xn = . n . ( ~) 1.36, nI X n (X,X)~-. 'mn=. ( ~ ) (1.46) (,) = . ' ( 1.36). 1.50 . 1.51. . (, ) C . , , . . : . Bolzano- Weierstrass . (.. , ' , , ' Riemann). Bolzano- Weierstrass ' . , . , ' , . . ( 5). 1.52. . ' (xn) . (nk) , (Xnk ) (xn ). , : -. , : -. , : -. .
20. 33. 26 1.53. . (, ) . (Xn ) ( ) ( ). . ( ~ ) ' (Xnk ) (Xn ) > . , ( ,) < ~ . (nk) , ko llko ~ , k;::: ko llk;::: llko ;::: . ' k;::: ko P(Xnk ,) < . 1imxn =. k k ( ?= ) , ' (xn ) (Xn ). 1.54. . (Xn ) (,) . (xn ) , (xnk ) (xn ) > , P(Xnk ,);::: k = 1,2,.... . (Xn ) , > , m ,
21. 34. 27 ( , ) > > , , ( ,) < , (f(xo ), f(x)) < . f , > > , f(S(xo ,)) C s(f(Xo ),). f 17 . 1.56. . () f: -+ , (,) (,), , ' (f(),f())=(,)= , , f. () , , (, ) . f: -+ . , > f('S(,I))={f()}CS(f(),). () , f: -IR C IR, . , IR , , . , , . , , ( 8), . 1.57 ( , , , ). 1.57. . (,), (,) f: -+ . : () f . () G , (G) . () F , (F) . () C J f(A )C~(A))- . () =:> () (G). G f(x) G, > S (),) C G. f , > f(S(x, )) C G. ' S(,) C (G). (G) . () =:> () Y,"""",F , , (Y,"""",F) J . (Y,"""",F) = ,"""", (F). ' ' (F) . () =:> () C . f(x) f(A) C f(A): C (1 ()). f(A) , , (f(A)) . ', 1.32, C(f()), f(A)Cf(A).
22. 35. 28 () ~ () f . > . > S(X() ,) C (S(f(),). = (S(f(),)) = { : (f(xo),f(x)) S( ,)("'-..)#0. 1.33 ('-)-. ' f(xo) f(X '-)-). f(X'-A)-)C(f(X'-A))-. f(xo) (f(X'-A))-. , 1.33, S(f(),)f('-)#0, ' (f(xo), f(x)) < , . f . . , , [ ], . , 1.57, ( 8). , ' . 1.58. . (, ), (,) [: --+ . : () f . () (xn) , limxn = limf(xn) = f(x). n n . () ~ () (Xn) , limxn = n (f(xn)) f(x). 1.54, (f(xnk )), f(xn), > cr(f(xnk),f(x))~ k = 1'2'.... c > f(S(,))CS(f(),). (nk,)~ k = 1,2,..., ( 1.53). () ~ () f . ' > , > f (S (, ))
23. 36. 29 ( , ) > > , X,XI , (,l)