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8/12/2019 Walter Rudin -
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2
:
LEADER BOOKS A.E.
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1 Rotman Joseph: Galois, xii, 185 , fi 2000
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Walter Rudin
fi :
K.
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: Principles
of Mathematical Analysis
: Walter Rudin
: Third Edition 1976,
McGrawHill
Book Co.Singapore
Copyright 1964, 1976: McGrawHill, Inc.
Copyright 2000 : Leader Books A.E. fi : .
fi
: .
:
:
1 : 2000
ISBN 9607901169
fi
LEADER BOOKS A.E.
. 17, fi,
115 21
T. : 64.52.825-64.50.048, Fax.: 64.49.924
http://www.leaderbooks.com, e-mail:[email protected]
fi
fi .
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POOO TOY
METAPATH
A ,
Leader Books, -
Principles of Mathematical Analysis Walter Rudin,
Wisconsin.
O Rudin A,
fi fi fi -
. Efi fi fi , Rudin
fi , Real and Complex
Analysis Functional Analysis. T
M. T
, fi
M A.
fi -
fi (fi ' )
fi fi
fi, - .
M-
vii
8/12/2019 Walter Rudin -
6/543
, fi , fi
( ). fi , fi
fi, M , fi,
fi
.
T fi
:
BELL, E. T.: O , Efi K (2fi), H 1992, 1993.
BOYER, C. B. MERZBACH, U.C.:H M,Efi . A. ( ), A 1997.
DAVIS, P. J. HERSH, R.: H , EfiT, A.
PIER, J. P.: Development of Mathematics 1900-1950, Birkhuserpublications, Basel, Switzerland 1994.
LORIA, G.: I M, Efi , A1971.
SMITH, D. E.: History of Mathematics, Dover publications (2 fi),New York.
E , fi The MacTutor
History of Mathematics archive, -
http:// www-history.mcs.st-and.ac.uk / history, M-
St. Andrews .
fi fi
fi,
fi . fi fi fi .
fi
viii
8/12/2019 Walter Rudin -
7/543
. E fi fi -
fi fi fi fi ,
. fi
fi ,
-
fi .
A -
I Nfi M
, -
I fi fi
I fi fi-
, I
I
N fi fi -
. ,
I M M -
K Jesper Ltzen
fi Johannes Mollerup.
, Leader Books
fi , fi
.
HMOENH K. TAIH
I 2000
ix
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POOO TOY
YPAEA
T fi fi -
fi A
M
E.
H 1 ,
,
. fi -
fi fi fi .
H fi (
fi) fi -
fi . , fi
fi
. fi fi,
fi
,
. ' fi , 1 . . M.: H , 1976, fi
fi McGraw-Hill.
xi
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fi
TOY METAPATH vii
TOY YPAEA xi
1
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
. . . . . . . . . . . . . . . . . . . . . . . 4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
. . . . . . . . . . . . . . . 12
fi 16
. . . . . . . . . . . . . . . . 17 . . . . . . . . . . . . . . . . . . . . . . . . . 22
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 37
, . . . . 37
. . . . . . . . . . . . . . . . . . . . . . . . . . 47
. . . . . . . . . . . . . . . . . . . . . . . . . . 56
. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
. . . . . . . . . . . . . . . . . . . . . . . . . 66
3 AKOOYIE KAI EIPE 75
. . . . . . . . . . . . . . . . . . . . . 75
xiii
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. . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Cauchy . . . . . . . . . . . . . . . . . . . . . . . . 82 fi . . . . . . . . . . . . . . . . . . . 88
. . . . . . . . . . . . . . 90
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
fi . . . . . . . . . . . . . . . . . . . . 96
fie . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
fi. . . . . . . . . . . . . . . 102
. . . . . . . . . . . . . . . . . . . . . . . . . . . 106
. . . . . . . . . . . . . . . . . . . . . . . . 108
fi . . . . . . . . . . . . . . . . . . . . . . . . . 110fi fi . . . . . . . . . . . . 111
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4 YNEXEIA 129
. . . . . . . . . . . . . . . . . . . . . . . . 130
. . . . . . . . . . . . . . . . . . . . . . . 132
. . . . . . . . . . . . . . . . . . . . . . 137
fi . . . . . . . . . . . . . . . . . . . 143
. . . . . . . . . . . . . . . . . . . . . . . . . . 144fi . . . . . . . . . . . . . . . . . . . . . . 146
fi fi . . . . . . . . . . . . . . . . 149
5 IAOPIH 159
. . . . . . . . . . . . 159
. . . . . . . . . . . . . . . . . . . . . . 164
. . . . . . . . . . . . . . . . . . . . . 166
fi L' Hospital . . . . . . . . . . . . . . . . . . . . 167
. . . . . . . . . . . . . . . . . . . 169 Taylor . . . . . . . . . . . . . . . . . . . . . . 170
. . . . . . . . . . . 171
xiv
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6 TO KATA RIEMANN-STIELTJES OOKHPMA 189
fi . . . . . . . . . . . . 1 90fi . . . . . . . . . . . . . . . . . . 201
. . . . . . . . . . . . . . . . . . 209
. . . . . . . . . . . 2 11
. . . . . . . . . . . . . . . . . . . . 212
7 AKOOYIE KAI EIPE YNAPTHEN 223
. . . . . . . . . . . . . . . 224
fi . . . . . . . . . . . . . . . . . . . . . . . 228
fi . . . . . . . . . . . . . . . 231
fi . . . . . . . . . . . . . 235fi . . . . . . . . . . . . . 236
. . . . . . . . . . . . . . 240
Stone Weierstrass . . . . . . . . . . . . . 246
8 OPIMENE EIIKOY TYOY YNAPTHEI 267
. . . . . . . . . . . . . . . . . . . . . . . . . . . 267
. . . . . . . . . . . . 276
. . . . . . . . . . . . . . . . . 2 82
fi 285 Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
. . . . . . . . . . . . . . . . . . . . . . . . 298
9 YNAPTHEI ON METABHTN 319
. . . . . . . . . . . . . . . . . . . 319
fi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
fi . . . . . . . . . . . . . . . . . . . . . 342
. . . . . . . . . . . . . 344
. . . . . . . . . . . . . 347
. . . . . . . . . . . . . . . . . . . . . 353 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
. . . . . . . . . . . . . . . . . . . 364
xv
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. . . . . . . . . . . . . . . . . . 367
10 OOKHPH IAOPIKN MOPN 381
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
. . . . . . . . . . . . . . . . . . . . 386
. . . . . . . . . . . . . . . . . . . . . 390
. . . . . . . . . . . . . . . . . . . . . . . . 391
. . . . . . . . . . . . . . . . . . . . . . . . 393
fi . . . . . . . . . . . . . . . . . . . . . 411
Stokes . . . . . . . . . . . . . . . . . . . . . . 421
. . . . . . . . . . . . . . . . . . 425
. . . . . . . . . . . . . . . . . . . . . . 433
11 H EPIA TOY LEBESGUE 463
. . . . . . . . . . . . . . . . . . . . . . . . 464
H Lebesgue . . . . . . . . . . . . . . . . 467
. . . . . . . . . . . . . . . . . . . . . . . . . . . 478
. . . . . . . . . . . . . . . . . . . . . 478
. . . . . . . . . . . . . . . . . . . . . . . . 482
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
Riemann . . . . . . . . . . . 495
. . . . . . . . . . . . . . 498
L2 . . . . . . . . . . . . . . . . . . . . . 499
515
xvi
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K 1
A (fi , , fi )
. , fi
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2
fi ( m/n, fim, n
n= 0). fi -
fi, . ( fi
fi 1.6 1.12). , fi
fi p p2 = 2. ( fi fi ). fi ,
fi fi .
,
1, 1, 4, 1, 414, 1, 4142, . . .
2. , fi fi
fi
2 , : fi
;
fi
fi fi .
1.1. fi
p2 = 2 (1)
fi fi fi p. E fi
p, fi p= m/n, fi m, n fi 2. fi, (1)
fi
m2 = 2n2. (2)
Afi fi m2, m, . (E m
fi, fi m 2 fi). , m 2 fi 4. fi fi fi (2) fi 4,
n2 . fi fi fi n .
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3
fi (1) fi
m, n , fi m, n., (1) , fi fi p.
, .
p p2 2. fi
fi fi fi fi fi
B fi fi fi fi fi
.
, p
q p < q p
q q < p.
fi, fi fi p >0 fi
q= p p2 2
p + 2 =2p + 2p + 2 . (3)
Tfi,
q2 2 = 2(p2 2)
(p + 2)2 . (4)
E p , fi p2 2 < 0, (3) fi q > p (4) fi q 2 0, (3) fi 0 < q < p (4) fi q 2 >2. , q B.
1.2. H fi
fi -
, fi fi fi
: r
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4
, ,
fi .
-
fi .
Ofi 1.3. E (
), fi x A fi x ( ) .
x / A fi x .
fi . fi fi
.
E , fi
fi fi ,
fi fi A B A. E, fi, , fi, fi fi, A
. fi A A .
A=
B fi A
B
A.
= B.
Ofi 1.4. fi 1,
Q.
E
Ofi 1.5. S . S ,
8/12/2019 Walter Rudin -
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5
.
(ii) Ex,y ,z S x< y y x x < y.
fi x y fix < yx= y, fi . ,
x y x > y.
Ofi 1.6. S
.
fi Q, fi r, s Qr
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6
T , infimum, fi
fi fi: fi
= infE
fi
> .
1.9.
() 1.1
fi Q. . -
, . fi ,
Q.
, :
fi fi fi
rr 0. fi A , Q.
() E =sup E, fi fi fi . , E1 r r
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7
1.9() fi Q fi
. fi -
fi
fi fi
.
1.11. fi S
fi fi B S, fi fi . L
. fi,
= supL
S fi = infB . , infB S.
Afi. fi B , L fi. L
fi fi y S fiy x, x B, x B L . , L . , L supremum S,
.E S < , fi ( fi 1.8)
L , fi fi / B. , x x B. , L .
E S < , fi / L , fi L .
A fi L L fi S > . , B, , fi > . Afi
fi= infB .
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8
Ofi 1.12. F ,
fi fi, fi (A), (), (D):
()
(A1) Ex,y F, fi x+y F.(A2) fi : x+y= y +x x,y F.(A3) fi : (x+ y)+z= x+ (y+ z) x,y ,z F.(A4) To F 0, 0 + x= x x F.(A5) x F x F
x+ (x) = 0.
()
(M1) Ex,y F, fi fi x y F.(M2) fi fi: x y= yx x,y F.(M3) fi fi: (x y)z= x(yz) -x,y ,z F.(M4) To F 1 1= 0 1x= x
x F.(M5) x F x=0 1/x F
x(1/x) = 1.(D) fi fi
x,y ,z F fi
x(y + z) = x y + x z.
1.13.
()
xy , xy
, x+y + z, x yz , x2, x3, 2x, 3x, . . .
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10
fi 1.15.
fi , x,y,z F:() E x= 0 x y= x z, fi y= z .() E x= 0 x y= x, fi y= 1.() E x= 0 x y= 1, fi y= 1/x.() E x= 0, fi 1/(1/x) = x.
fi fi 1.14
.
fi 1.16. fi -
, x,y,z
F.
() 0x= 0.() E x= 0 y= 0, fi x y= 0.() (x)y= (x y) = x(y).() (x)(y) = x y.
Afi. 0x+ 0x= (0 + 0)x= 0x. , 1.14() fi fi 0x= 0 fi ().
E , fi x= 0, y= 0, x y= 0. Tfi, () fi
1 =
1y
1x
x y=
1y
1x
0 = 0,
fi . , ().
fi () fi
(x)y + x y= (x+ x)y= 0y= 0,
fi 1.14(). fi
fi.
,
(x)(y) = [x(y)] = [(x y)] = x y, () 1.14().
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11
Ofi 1.17. F,
, :(i) Ex,y,z F y 0 y >0, fi x y >0.O x fi fi x > 0. O x
fi fi x 0 fi x 0 y 0.() E 0< x < y, fi 0 < 1/y 0, fi 0= x+ x >x+ 0 x yy= 0, x(zy) >0
x z= x(z y ) + x y >0 + x y= x y.
() fi (), () 1.16() fi
[x(z y )] = (x)(z y ) >0,
fi fi x(z y )
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12
() Ex >0, fi (ii) 1.17 fi x2 >0.
E x < 0, fix > 0, (x)2
> 0. , fi x2
= (x)2
, fi 1.16(). fi 12 = 1, fi 1 > 0.
() E y >0 v 0, fi yv 0. , fi y (1/y)= 1 >0, fi 1/y > 0. , 1/x > 0. E
fi x < y fi (1/x) (1/y), fi 1/y
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13
() E x,y R x > 0, fi fi
fi n nx > y.
() E x,y R x < y, fi p Q x < p < y. () fi R.
() fiQ fiR :
fi fi.
Afi.
() nx,fi n
fi . E () , fi
y . , R.
=sup A. Efi x >0, x < x . , x 0 , fi (), fi fin
n(y x) >1.
Efi (), m 1, m2m 1 >n x m 2 > nx. Tfi,
m2
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14
n-
. fi fi R (fi
2).
1.21. fi fi fi x fi
fi n fi fi fi
fi y yn = x.
fi fi n
x x1/n .
Afi. fi y , fi y1,y2 -
0 < y1 < y2, fi yn1 < yn2 . t tn < x.
Et= x/(1 +x), fi 0< t x, t / E. , 1 + x . , 1.19
y= supE.
fi yn = x fi fi fiyn < x yn > x .
Afi fibn an = (b a)(bn1 + bn2a + + ban2 + an1) fi
bn an < (b a)nbn1,
fia, b 0 < a
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15
, (y+ h)n < x y+ h E. E fi y+ h > y, fi
fi fi y . fi yn >x.
k= yn x
ny n1 .
Tfi, 0< k< y. t y k, fi fi
yn tn yn (y k)n x t / E. Afi fi fi y k . y
k< y, fi fi y
.
, fi yn = x fi .
fi. E a, b n
fi fi , fi
(ab)1/n = a1/n b1/n.
Afi. =
a1/n,=
b1/n . Tfi,
ab= n n = ()n ,
fi fi fi. ( (M2) fi
1.12). fi fi 1.21 fi
(ab)1/n = = a1/nb1/n .
1.22 . fi -
. fi fi x > 0. n0
fi n0 x. ( fi n0
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16
fi R). n0, n1, . . . , nk1,
n k fi n0 +
n1
10+ + nk
10k x.
n0 +n1
10+ + nk
10k (k= 0, 1, 2, . . . ) . (5)
fi,x= supE. fi x
n0, n1n2n3 . (6)
fi, fi (6),
(5) (6) fi supE.
fi fi
, .
Ofi 1.23. fi
fi R
+ . R
< x < +, x R.
fi + fi
fi .
E , fi
, , fi fi fi sup E= +.
fi .
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17
fi -
, fi fi :
() E x fi fi, fi
x+ = +, x = , x+ =x
= 0.
() E x>0, fi x (+) = +, x () = .() E x
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19
(D)
x(y + z) = (a, b)(c + e, d+ f)= (ac + ae bd b f, ad+ a f+ bc + be)= (ac bd, ad+ bc) + (ae b f, a f+ be)= x y + x z.
H fi .
1.26. a, b
fi
(a, 0) + (b, 0) = (a + b, 0), (a, 0)(b, 0) = (ab, 0). fi fi.
1.26 fi (a, 0)
fi
a. , (a, 0) a.
fi
.
fi
1. fi fi (a, b) fia + bi .
Ofi 1.27. i= (0, 1).
1.28. I fi i 2 = 1.
Afi. i 2 = (0, 1)(0, 1) = (1, 0) = 1.
1.29. E a, b , fi (a, b) = a +bi .Afi. a + bi= (a, 0) + (b, 0)(0, 1) = (a, 0) + (0, b) = (a, b).
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20
Ofi 1.30. E a, b z= a+ bi , fi
fi fiz= a bi o z . a b fi fi
z .
K
a= Re(z), b= Im(z).
1.31. E z, w , fi
() z + w=z + w,() zw
=z
w,
() z + z= 2Re(z),z z= 2iIm(z),() o zz fi fi . I, fi , fi
z= 0.
Afi. fi (), (), () fi
fi. (), z = a+ bi , fi a, b , fi z z= a2 + b2.
Ofi 1.32. fi |z|fi z z z. , |z| = (zz)1/2.
( fi) |z| fi 1.21 () 1.31.
fi x fi fi, fi x= x |x| =
x2. , |x| = xx 0 |x| = xx 0, fi z= 0, |0| = 0.()|z| = |z|.
()|zw| = |z||w|.()|Re(z)| |z|.()|z + w| |z| + |w|.
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21
Afi. fi () () fi.
z= a + bi ,w= c + di , fia, b, c, d . fi,|zw|2 = (ac bd)2 + (ad+ bc)2 = (a2 + b2)(c2 + d2) = |z|2|w|2,
|zw|2 = (|z||w|)2. T, () fi fi fi, 1.21.
fi (), fi a 2 a2 + b2
|a| =
a2
a2 + b2.
fi (), fi z w z w
z w + zw= 2Re(zw). ,
|z + w|2 = (z + w)(z + w)= zz + zw + wz + ww= |z|2 + 2Re(zw) + |w|2
|z|2 + 2|zw| + |w|2
= |z|2 + 2|z||w| + |w|2
= (|z| + |w|)2.
, () .
fi 1.34. Ex1, . . . ,xn , fi 1
nj=1
xj= x1 + x2 + + xn .
H fi fi fi,
fi Schwarz2.
1 . . M.: fi fi fi : x1, . . . ,xn , fi
nj=1xj= x1 x2 . . . xn.2 . . M.: Karl Hermann Amandus Schwarz (1843-1921). fi fi. T
A .
T 1860 Schwarz B X.
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22
1.35. E a1, . . . , an b1, . . . , bn ,
fi n
j=1aj bj
2
n
j=1|aj |2
nj=1
|bj |2.
Afi. A= nj=1 |aj |2, B= nj=1 |bj |2, C= nj=1aj bj . EB = 0, fi b1 = = bn = 0 . fi B >0. 1.31
nj=1
|Ba j C bj |2 =n
j=1(Ba j C bj )(Ba j Cbj )
= B2
nj=1 |aj |
2
B Cn
j=1 aj bj BC
nj=1 a
j bj+ |C|2
nj=1 |bj |
2
= B2A B |C|2
= B(A B |C|2).
fi fi fi, -
fi
B(A B |C|2) 0.Efi B > 0, fi A B |C|2 0. , fi.
EYKEIEIOI
fi fi Karl Weierstrass (1815-1897) Ernst
Eduard Kummer (1810-1893), M. O Schwarz
fi Kummer fi fi .
fi 1864. T 1866 Schwarz
. K 1867 Halle
. T 1869 Schwarz
fi Z. K 1875,
Gttingen. T 1892 Schwarz Weierstrass B, fi 1917.
O Schwarz fi M. E
fi .
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23
Ofi 1.36. fi fik
Rk
fi k-
x = (x1, . . . ,xk),
fix1, . . . ,xk , -
x. T Rk ,
fi k>1. . E y=(y1, . . . ,yk) a fi fi, fi
x + y = (x1 +y1, . . . ,xk+yk),ax
=(ax1, . . . , axk).
, x + y Rk ax Rk. M fi fi fi fi
fi (fi ). -
fi, fi fi ( fi
, -
) Rk fi
. fi Rk (
fi ) 0,
fi 0.
, fifi fi( fi
fi) xy
x y =k
i=1xiyi
x
|x| = (x x)1/2 =
ni=1
x2i
1/2.
H o ( fi Rk
fi fi ) E
k-.
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24
1.37. Y fi x, y, z Rk fi a -
fi fi . fi:() |x| 0.()|x| = 0 fi x = 0.()|ax| = |a||x|.()|x y| |x||y|.()|x + y| |x| + |y|.()|x z| |x y| + |y z|.
Afi. (), () () ()
fi Schwarz. Afi () fi
|x + y|2 = (x + y) (x + y)= x x + 2x y + y y |x|2 + 2|x||y| + |y|2
= (|x| + |y|)2,
' fi fi (). , () fi
() x x y y y z.
1.38. (), () () 1.37 - Rk fi (
2).
R1 ( ) -
. , R2 fi
( 1.24 1.36). -
, fi
.
AP
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25
fi 1.19,
Rfi Q. .
1. R Q,
. , ,
Q fi:
() To fi = Q.
(I) E p ,q Q q < p, fiq .
(II) E p , fi p < r r .
p, q, r, . . . , , , . . . .
fi (III) fi .
T (II) fi,
:
E p q / , fi p < q .Er / r
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, R .
B 3. R fi
.
fi , fi
A R. fi A.
fi A. M , fi p fi p A. fi R fi= sup A.
Efi A fi, 0 A. To 0 fi.fi0 , fi. Kfi, (fi , A) = Q. , fi (). fi (II) (III), p . fi, p 1 1 A. E q < p, fi q 1 q . Afi (II). E r 1 r > p, fir (fi1 ). Afi (III).
, R.fi fi , A. fi < . fi, s s / . Efi s ,
fi s A. , <
.fi = supA.
4. E , R, fi + r+ s, fi r s .
0 .
fi fi .
R( fi 1.12), 0
0.
(A1) fi
+ . fi
+ fi. r / , s / . Tfi, r s , fi r+ s > r+ s. , r+ s / + . , + fi ().
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p + . Tfi, p= r+ s r s . E q < p,
fi q s < r q s . , q= (q s) + s + ., (II). t t > r. Tfi, p < t+ st+ s + . M fi (III).
(A2) + r+ s, fir s . , + s + r, fis r . fir+ s= s + r, r, s Q, fi + = + .
(A3) , fi fi
fi Q.
(A4) E r
s
0, fi r+
s < r, r+
s
. ,
+ 0 . fi p .A r r > p. fi, p r 0 p= r+ (p r) + 0. , + 0. K' fi fi + 0 = .
(A5) R. A p fi fi:
fi fi r >0 p r / ., fi fi, fi p, o
.
fi
R
+
=0.
E s / p= s 1, fip 1 / p ., fi. E q , fi q / = Q., ().
p r >0, p r / . E q < p,fi q r > p r, q r / . ,q (II). t= p+ (r/2). Tfi, t > p t (r/2)= p r / .E,t . K , (III).
fi R.E r s , fis / r
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fin n w (n+ 1)w / . ( fi fi
fi Q.) p= (n+ 2)w. fi, p ,fi p w / . E
v= nw + p + .
, 0 + . fi fi + = 0., , fi , .
5. fi fi, fi
4, () 1.12, fi fi
1.14 R fi 1.17:
E , , R < , fi + < + ., fi fi R fi +
+ . E + = + , fi fi (fi 1.14) fi= , fi .
, fi >0 fi 0.
E , R+, fi fi p p r s, r r >0 s s >0.
1 q q 0 >0, fi >0.
7. fi
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0 = 0= 0
=
()() 0. fi, = (+ )+() (fi fi fi fi R+)
= (+ ) + ().
, () = (). ,
+
=(
+ ).
fi fi .
fi fi R
fi .
B 8. r Q r p p
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fi, p (r+ s). fi, p < r+ s. t
2t= r+ sp. r= r t, s= s t.
fi,r r, s s p= r + s. , p r + s.fi (). () .
E r
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Knopp fi
R fi Q. fi Hewitt Stromberg
, fi fi fi
Cauchy fi . (
3).
R
Dedekind3. H Rfi Q Cauchy
o Cantor.
1872.
3 . . M.: Julius Wilhelm Richard Dedekind (1831-1916). fi
fi. E A. M , fi
fi , Dedekind, fi
. E,
, fi.
fi fi fi .
, Dedekind , Braunschweich. K 1848
K Caroline,
M. E Gttingen, 1850, fi
fi Carl Friedrich Gauss (1777-1855), Wilhelm Eduard Weber (1804-1891) Moritz
Abraham Stern (1807-1894). O Dedekind fi fi Gauss,
A, 1852. Gttingen 1855
fi fi Z,
. O Dedekind 1862 Braunschweich
, fi
.
O Dedekind fi fi
fi fi fi . Xfi
fi fi fi M
Dedekind 4 1899. O
fi fi fi fi fi Georg Cantor(1845-1918). O Dedekind, fi
, fi
Cantor. O Dedekind fi Bernhard Riemann (1826-1866).
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AKHEI
fi fi, fi
fi .
1. E r fi fi (r= 0) x fi (x= 0), fi fi r+ xr x .
2. fi fi fi
12.
3. fi 1.15.
4. E fi fi fi.
fi E E.
fi .
5. A A fi
. A x, fi x A.A fi
infA=
sup(
A).
6. b > 1.
() E m, n,p, q n >0, q >0
r= m/n= p/q, fi fi
(bm )1/n = (bp)1/q .
M fi, fi fibr = (bm )1/n .() fibr+s = brbs , fi r, s .() E x fi fi, fi B(x)
bt,fi t fi fi t x. fi
br = supB(r),
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r fi fi. , fi fi
bx = supB(x),
fi fi x.
() fi bx+y = bxby , -x,y .
7. b > 1,y > 0. fi -
fi fi x bx = y, fi. (x y b.)
() fi fi n fibn
1 n(b 1).() fi fi b 1 n(b1/n 1).() E t >1 n > (b 1)/(t 1), fib1/n y, fi bw(1/n) > y,
fi fin .
() w bw < y. fi
x
=supA, fibx
=y.
() o fi fi fi x fi.
8. A fi
.
Yfi:T 1 .
9. fi z= a+ bi, w= c + di . z < w fi a < c a= c b < d. A fi . (
, fi. fi fi
;
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10. fi z= a + bi, w= u + vi fi
a= |w| + u
2
1/2, b=
|w| u2
1/2.
fiz 2 = wv 0 fi(z)2 = wv 0. fi fi fi ( !)
.
11. E z fi fi, fi fi
r0 fi fi w |w| =1 z= rw. r, w fi z ;
12. Ez 1, . . . ,zn , fi fi
|z1 + + zn| |z1| + + |zn|.
13. Ex,y , fi fi
||x| |y| | |xy |.
14. Ez fi fi |z| =1, z z=1, fi
|1
+z
|2
+ |1
z
|2.
15. fi fi fi
Schwarz;
16. fik 3, fix, y Rk |x y| = d>0 fir >0. fi:
() E 2r >d, fi z Rk
|z x| = |z y| = r.
() E 2r
=d, fi z fi .
() E 2r
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17. A fi
|x + y|2 + |x y|2 = 2|x|2 + 2|y|2,
x, y Rk. fi fi , fi fi.
18. E k 2 x Rk, fi fi y Rk y = 0 x y = 0. fi k= 1;
19. fia, b Rk. c Rk r >0
|x a| = 2|x b|
fi |x c| = r.(:3c = 4b a, 3r= 2|b a|.)
20. , fi fi (III)
. -
. fi
fi fi fi
(A1) (A4) ( fi fi!), fi (A5).
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K 2
, -
fi -
.
Ofi 2.1. , , fi x ,
fi, , f(x).
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38
fi, f fi (
fi ). f( fi f ). f(x),
fi x A, f fi f f.
Ofi 2.2. , f fi .
E E A, fi f(E) f(x) x E. O fi fi f. , f(A) f.
fi f(A)
B. H f fi
fi f(A) = B. (M fi, f fi f .
E E B, fi f1(E) x A f(x) E. fi fi f. E y , fi f1(y) x A f(x)= y. H f 1-1 (--)fi fi f1(y)
. : f 1-1
fi fi f(x1) = f(x2) x1,x2 A x1=x2.
( fix1= x2 fi x1,x2 . ,x1= x2.)
Ofi 2.3. fi 1-1
fi fi fi fi, ,
fi 1-1 fi 1. fi
fi A B. fi: : A.
1
. . M.: M ffi A B fi 1-1 B .
gfi B A , f1, fig(f(x)) = xx A f(g(y)) = y y B. H g fi f.
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: E B, fi B A.
: E A B B C, fi A C.K fi .
Ofi 2.4. fi fi n Jn
1, 2, . . . , n. , -
J fi . E
, fi :
() fi Jn, fi fin ( fi ).
() fi .
() fi A J.() Y fi
.
() fi -
.
.
fi A B fi fi . ,
, ,
1-1 .
2.5. fi . fi,
. fi
fi J:
A: 0, 1, 1, 2, 2, 3, 3, . . .J : 1, 2, 3, 4, 5, 6, 7, . . .
fi
f, 1-1 :
f(n) =
n2 n fi,
n 12 n fi fi.
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2.6.
fi . , fi fi , fi 2.5, fi J
A.
fi, fi2.4()
fi fi: fi
fi .
Ofi 2.7. fi f,
J . E f(n)= xn , n J, fi f {xn}(n= 1, 2, 3, . . . ) {xn}, x1,x2,x3, . . . . f , xn n J, fi . E xn A n J, fi {xn} (n=1, 2, 3, . . . ) .
fi fi x1,x2,x3, . . . -
.
Efi 1-1
J, -
fi.
B fi, fi fi
fi .
fi J
fi fi ,
0 1.
2.8. fi fi
.
Afi. fi A. -
{xn} (n= 1, 2, 3, . . . ) - . {nk}(k= 1, 2, 3, . . . ) :
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n1 fi fi fi n1 xn1 E.
n1, . . . , nk1, nk fi fi fi fi
nk1 xnk E. fi f fi J E f(k)= xnk (k=
1, 2, 3, . . . ), 1-1 J.
fi fi,
fi fi :
fi.
Ofi 2.9. fi
,
E.
E, fi A, {E}( A) {E}. A fi fi fi fi
fi.
{E}( A) S o fi: x
S fi x
E
A.
fi
S=A
E. (1)
E fi 1, 2, . . . , n, fi -
S=n
m=1Em , (2)
S= E1 E2 En . (3)
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E , fi
S=
m=1Em, (4)
(4) fi - fi + fi 1.23.
{E} ( A) P o : x P fi x E A. fi
P=A
E, (5)
P=n
m=1Em= E1 E2 En , (6)
P=
m=1Em , (7)
fi .
A,B fi fi A fi. , ,
.
2.10.
() fi 1 fi 1, 2, 3
2fi 2, 3, 4. fi, E1 E2 fi 1, 2, 3, 4 E1
E2fi 2, 3.
() x 0 < x 1. x A, Ex y 0< y < x. fi,
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43
(i) x Ez fi 0 < x z 1.
(ii)
xA Ex= E1.(iii) To
xA Ex fi.
(i) (ii) . fi (iii) fi
fi fi y y > 0 fi y / Ex x < y. ,y / xA Ex. 2.11. fi
. -
fi, fi
fi fi
.
fi fi fi -
:
A B= BA, A B= BA. (8)
A (B C) = (A B ) C, A (B C) = (A B ) C. (9)
fi fi (3) (6).
fi fi:
A (B C) = (A B ) (A C). (10)
fi, E
(10) F .
fix E. fi,x A x B C, x B x C( ). ,x A B x A C, x F. ,E F.
, fix F. Tfi,x A B x A C. ,x A x B C. , x E. , F E.
fi fi E= F.
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fi ,
.
A B , (11)
A B A. (12)
E 0 fi , fi
A 0 = A, A 0 = 0. (13)
E A B, fi
A B= B, A B= A. (14)
2.12. Y fi{En} (n= 1, 2, 3, . . .) - fi.
S=
n=1En. (15)
Tfi, S .
Afi. n {xnk} (k=1, 2, 3, . . . )
x11 x12 x13 x14 . . .
x21 x22 x23 x24 . . .
x31 x32 x33 x34 . . .
x41 x42 x43 x44 . . .
. . . . . . . . . . . . . . . . . . . . .
(16)
En n.
fi S.
x11: x21,x12: x31,x22,x13: x41,x32,x23,x14: . . . (17)
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E fi n ,
fi (17). , fi S T, fi fi S ( 2.8). fi
E1 S 1 , fi S .
fi. fi
A B . fi,
T=A
B
.
fi fi fi
(15).
2.13. Bn
n- (a1, . . . , an ) ak A (k= 1, 2, . . . , n). (T a1, . . . , an .) fi, Bn
.
Afi. fi 1 . fi
Bn1(n= 2, 3, 4, . . . ) . Bn
(b, a) (b Bn1, a A). (18)
b n1, (b, a)
. , Bn
fi. fi 2.12
fi Bn .
.
fi. .
Afi. fi 2.13 n= 2, fi fi fi r b/a, fi a, b .
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(a, b),
b/a, .
fi, fi
( 2).
, fi fi fi ,
.
2.14. Y fi
fi 0 1. fi, .
fi 1, 0, 0, 1, 0, 1, 1, 1, . . . .
Afi. . fi
fi s1, s2, s3, . . . .
s : n fi sn 1, fi
n fi s 0 fi . fi, s
fi . , s / E. s A, .
, fi
. , (
.)
fi fi
Cantor 2 Cantor:
2 . . M.: Georg Cantor (1845-1918). E fi fi ,
P. O Cantor fi.
O Cantor M,
Z, 1862, M, B,
1863. E Ernst Kummer (1810-
1893), Karl Weierstrass (1815-1897) fi
( fi fi ) Leopold Kronecker (1823-1891).
B 1867. T A. E, fi
Weierstrass, T .
O Cantor Halle 1869, fi
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s1, s2, s3, . . . fi (16), fi
.
( 2 fi 10)
fi 2.14 fi
. fi
fi 2.43 fi fi.
Ofi 2.15. X, ,
fi fi (p, q)
X fi fi d(p, q),
fi pfi q , p, q, r X :()d(p, q) >0, p= q . ,d(p,p) = 0.()d(p, q) = d(q,p).()d(p, q) d(p, r) + d(r, q). fi -
.
2.16.
fi Rk, R1 (
1913.
B, fi , fi fi Kronecker
, fi fi,
, Cantor.
O Cantor 1874 fi fi,
fi
M.
O Cantor . fi fi , fi fi
fi fi . T fi
Halle, fi .
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) R2 ( fi ). fi d Rk
d(x, y) = |x y| (x, y Rk). (19)
1.37, 2.15 -
fi (19).
fi fi Yfi
fi .
fi fi () () 2.15
p, q, r X, fi p, q, r ., fi E fi .
C(K) L2(),
7 11 .
Ofi 2.17. fi fi (a, b)
x a < x 0, fi () x r
y
Rk |y
x|
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fix, y E R1 0 < 0 R1 |y x| < r,|z x| < r 0<
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() E fi
fi fi Mq Xd(p, q) < M, p E.() E fi fi
X fi E E( ).
fi R 1
R2 .
2.19. X fi .
Afi. E= Nr(p)(p X, r >0). q . fi, fi fih
d(p, q) = r h.
s d(q, s)
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fi.
.
2.21. fi R2:
() z |z|
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fi. fi,
A
E
c
=A
Ec. (20)
Afi. A B (20). E x A,fi x / A E x / E A., x Ec A, x
AE
c.
, A B.fi, x B, fi x Ec A,
x / E A. , x
A E
c.
, B
A.
, A= B.
2.23. E fi fi
fi.
Afi. , fi Ec fi. x E.fi,x / Ec, x fi Ec. , Nx Ec N fi. fi fiN E. , x fi E. K , Efi.
fi, fi E fi. x fi
Ec. Tfi, x Ec, fi
fi x fi E. fi E
fi, fi x Ec. , Ec fi.
fi. F fi fi -
fi.
2.24.
()
{G
} (
A) fi, AG fi.
() {F} ( A) fi,
AF fi.
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() G1, . . . , Gn fi,
n
i=1G i fi.() F1, . . . ,Fn fi,
n
i=1 Fi fi.
Afi. G= AG . E x G, fi x G, . fi x fi G, fi
G . G fi. fi ().
fi 2.22, fiA
F
c= A
Fc (21)
Fc fi , 2.23.
, () fi (21) fi, AF fi.
, = ni=1G i . x H Nix ri Ni G i , i= 1, 2, . . . , n.
r= min{r1, . . . , rn}
N x r. fi, N Gi i=
1, 2, . . . , n. , N
H H fi.
, () fi (): n
i=1Fi
c=
ni=1
Fci .
2.25. () () ,
. -
fi, G n fi 1n
, 1n (n=
1, 2, 3, . . . ).
Tfi, Gn fi R1 n=1, 2, 3, . . . . , G= i=1G i fi fi ( 0) fi R1.
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, fi -
fi . , fi fi .
Ofi 2.26. X fi E X. E E , fi (
(fi) ) E= EE.
2.27. E X fi E X, fi() E fi.
() E=
E fi fi.
() E F fi F X E F.
fi () () fi E fifi
.
Afi.
() E p X p / E, fi p E fi E. , p E. ,
E fi, E fi.
() E E
=E, fi () fi E fi. E E
fi, fi E E( 2.18() 2.26) E= E.
() E F fi E F, fi F F, E F. , E F.
2.28. Y fi fi
. E y= supE, fi y E. (y E, fi).
fi fi 1.9.
Afi. E y E, fi y E. fi fi y / E. fi fih h >0 x E y h < x < y,
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y h . , y fi
, fi fi y E. 2.29. fiE Y X, fi Xfi . fi E fi X fi
p E fi fi r q Xd(p, q)
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Ofi 2.31. fi fi fiE fi
X {G}( A) fi E AG .Ofi 2.32. Kfi
fi K .
, : {G} ( A) K, fi 1, . . . , n
K G 1 Gn .
fi A, -
( 4).
fi . fi
2.41
fi Rk.
fi 2.29 fi E Y X, fi fi fi
. , fi fi fi
. .
, fi ,
fi . ,
fi 2.32.
2.33. fi K Y X. fi, fi .
fi ,
, .
, fi ( fi fi fi ),
fi .
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Afi. fi K .
{V} ( A) Yfi AV. 2.30, fi{G}( A), , V= Y G A. Efi K , fi
K G 1 Gn (22)
1, . . . , n . fi K Y, (22) fi
K V1 Vn . (23)
Afi fi .fi, fi .
{G} ( A) X fi AG . V= Y G A.fi, (23) 1, . . . , n . fi V G A, (23) (22).
fi fi.
2.34. K fi
fi.
Afi. Y fi K fi -
X. fi fi
X.
p X p / K. E q K, fi Vq p Wq q fi fi
12 d(p, q)(
fi 2.18()). fi K , q1, . . . , qn K
K Wq1 Wqn= W.
E V= Vq1 Vqn , fi V p W. ,V Kc, p fi Kc.M fi .
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2.35. K fi fi fi
fi .
Afi. fi F K X, fi F fi ( ) K . {V} ( A) F. E Fc {V}( A), fi K. fi ,
K, F.
E Fc , fi fi
F. , fi
{V} ( A) F.
fi. E F fi , fi F K .
Afi. 2.24() 2.34 fi F K fi. fiF K K, 2.35 fi F K.
2.36. E
{K
} (
A) -
fi fi X fi fi
{K} ( A) , fi A K fi.
Afi. K1{K}( A) G= Kc( A). fi K1 K. Tfi, {G} ( A) K1.fi K1 , 1, . . . , n
K1 G 1 Gn . fi fi
K1 K1 Kn
fi, fi fi .
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fi. E {Kn} (n = 1, 2, 3, . . .)
fi fi Kn+1 Kn (n =1, 2, 3, . . .), fi
n=1Kn fi.
2.37. E E fi
fi K, fi E fi K.
Afi. E K fi E, fi
q K Vq E( q, q E). fi {Vq}(q K) E K, fiE
K. fi fi fi K.
2.38. E {In} (n= 1, 2, 3, . . .) R1 In+1 In (n=1, 2, 3, . . .), fi
n=1 In
fi.
Afi. fi In= [an , bn] n= 1, 2, 3, . . . . E an , fi n= 1, 2, 3, . . . . Tfi, E fi (fi b1). A x= sup E. E m, n , fi
an am+n bm+n bm,
x bm fi fi m. fi am x, fi fi m, fi x Im , m= 1, 2, 3, . . . .
2.39. Y fi k fi fi .
E{In} (n= 1, 2, 3, . . .) k- In+1 In(n= 1, 2, 3, . . .), fi
n=1 In fi.
Afi. fi In fi x = (x1, . . . ,xk)
an,j xj bn,j (1 j k, n= 1, 2, 3, . . . )
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In,j = [an,j , bn,j ]. j , {n,j}
(n= 1, 2, 3, . . . ) 2.38. -, xj (1 j k),
an,j xj bn,j (1 j k, n= 1, 2, 3, . . . ) .
x = (x1, . . . ,xk) fi x In n =1, 2, 3, . . . . M fi .
2.40. k- .
Afi. k- I, fi x=(x1, . . . ,xk) aj xj bj (1 j k).
=
kj=1
(bj aj )21/2
.
Tfi, |x y| x, y I., , fi
{G} ( A) I I. cj= (aj +bj )/2 (1 j k). fi, [aj , cj ] [cj , bj ] 2k k-Qi (i
=1, 2, . . . 2k),
. fi ,
I1, fi
{G} ( A) ( ). A, I1 fi
. fi, k-{In} (n =1, 2, 3, . . . ) fi fi:
() I3 I2 I1 I.() To In fi {G}
(
A).
() Ex, y In, fi |x y| 2n (n= 1, 2, 3, . . . ). () 2.39, x
In. A x G. fi
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61
G fi, r >0 |y x| < r, fiyG.
E fi fi n 2n < r (fi n fi , fi
2n /r fi fi n, fi fi R1), fi () fi In G, ().
fi fi.
() ()
Heine3 Borel4.
2.41. E E Rk fi -
fi , fi fi :
3 . . M.: Eduard Heine (1821-1881). fi fi,
A. Y Carl Friedrich Gauss (1777-1855)
Gustav Peter Lejeune Dirichlet (1805-1859).4 . . M.: Felix douard Justin mile Borel (1871-1956). fi
A M (
1898). H , Henri Lon Lebesgue (1875-1941) Ren
Louis Baire (1874-1932), fi A
M. E,
. Efi , fi
.
O Borel Charles Hermite (1822-1901). K
cole Normale Suprieure , 1896. K
1909,
fi, fi. T 1918, Borel
fi ,
.
Efi fi , Borel -
fi. K , B
A, fi 1924 1936, Yfi N, fi
1925 1940. T 1940 Borel fi fi fi
(fi) Vichy fi fi . K
A N. T 1945 M A 1950 M fi T.
fi , Borel 1955 Xfi M E
K E (CNRS) .
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62
() E fi .
() E .() E fi .
Afi. E (), fi E I k- I () fi 2.40 2.35. 2.37 fi
() (). fi () fi
().
E E , fi E xn (n =1, 2, 3, . . . )
|xn| >n (n= 1, 2, 3, . . . ) .
fi fi
Rk . K , () fi
.
E E fi, fi x0 Rk fi E E. n= 1, 2, 3, . . . , xn E |xn x0|
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63
fi () () -
fi ( 26). , () - () (). 16. ,
L2,
11.
2.42 (Weierstrass5).
5 . . M.: Karl Wilhelm Theodor Weierstrass (1815-1897). K fi
fi. E A .
H M.
O Weierstrass fi , fi fi . O
fi Weierstrass fi , , Weierstrass fi N
Bfi 1834. M N, Weierstrass -
fi , fi fi ,
M,
N. K fi , Weierstrass
A Mnster
. Mnster, Weierstrass fi fi Christoph
Gudermann (1798-1851), fi fi Weierstrass.
O Weierstrass 1841, ,
, fi M. O Gudermann , fi
, Weierstrass . O Weierstrass
.
fi fi . T -
Knigsberg, , Weierstrass
1855. O Weierstrass 1856
M B B .
, Weierstrass M
B A E B-
. Afi fi , Weierstrass fi fi fi
fi, fi fi .
O Weierstrass B
fi . Y . M
Hermann Amandus Schwarz (1843-1921), Gsta Magnus Mittag-Leffler(1846-1927), Sonja Kowalewski (1850-1891) ( ), Immanuel
Lazarus Fuchs (1833-1902), Georg Ferdinand Frobenius (1849-1917), Carl David Runge (1856-
1927), Wilhelm Karl Joseph Killing (1847-1923) Hans von Mangoldt (1824-1868).
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64
Rk fi Rk.
Afi. E Rk.
, fi k- I Rk. 2.40, I E fi
I, 2.37.
2.43. P fi Rk.
fi, P .
Afi. fi P , .
fi P . P
x1, x2, x3, . . . . {Vn} (n =1, 2, 3, . . . ) :
V1 x1. E V1 r > 0, fi
V1 V1 y Rk |y x1| r. fi fi fi n Vn -
Vn P fi. fi
P fi P, Vn+1 (i)Vn+1 Vn , (ii) xn / Vn+1, (iii) Vn+1P fi. (iii), Vn+1 fi .
fi fi n Kn= Vn P . fi Vn fi , . fi xn / Kn+1, P
n=1Kn . fi Kn P , fi
n=1 Kn fi. , Kn fi, (iii),
Kn+1 Kn (n= 1, 2, 3 . . . ), (i). fi fi 2.36.
fi. [a, b] (a < b) . -
, .
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65
2.44 Cantor. To fi
fi R1
.
A E0 [0, 1]. A fi fi
13 ,
23
.
A E1 0,
1
3
,
2
3, 1
.
A fi . A
E2
0,
1
9
,2
9 ,3
9
,6
9 ,7
9
,8
9 , 1
.
fi fi, -
fi En(n= 1, 2, 3, . . . ) fi:() E3 E2 E1.() T En 2n , 3n .
T
P=
n=1En
Cantor. , P
2.36 fi fi.
E fi 3k+ 1
3m ,
3k+ 13m
(24)
P , fik, m . E
3m R.Afi. an= cnzn (n= 0, 1, 2, . . . ) fi :
limsupn
n
|an| = |z| limsupn
n
|cn| =|z|R
.
: R
n=0cnzn.
3.40.
() H
n=1nnzn R= 0.
() H
n=1zn
n! R= +. ( fi fi.)
() H
n=0zn R= 1. E|z| = 1, fi
, fi {zn} (n= 0, 1, 2, . . . ) 0n .
() H
n=1zn
n R= 1. A z= 1. fi fiz |z| = 1 . (O fi 3.44.)
() H n=1 znn2 R=1.
fi fi z|z| =1, , fi|zn /n2| = 1/n2 n .
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108
APOIH KATA MEPH
3.41. {an}, {bn} (n= 0, 1, 2, . . .),
An=n
k=0ak,
fi n 0 A1= 0. E 0 p q, fi fi
qn=p
anbn=q1n=p
An (bn bn+1) + Aq bq Ap1bp. (20)
Afi. E
qn=p
an bn=q
n=p(AnAn1)bn=
qn=p
An bnq1
n=p1An bn+1
(20).
O (20), fi ,
n=1an bn ,
fi {bn} (n= 1, 2, 3, . . . ) fi. .
3.42. Y fi:
() T An (n= 1, 2, 3, . . .)
n=1an -
.
() b1 b2 b3 .() limnbn= 0.Tfi, n=1an bn .
Afi. E fi M |An| M n. > 0, fi N bN (/2M).
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AKOOYIE KAI EIPE 109
p, q N p q
qn=p
an bn
=
q1n=p
An(bn bn+1) + Aq bq Ap1bp
M
q1n=p
(bn bn+1) + bq+ bp
= 2Mbp 2MbN .
T, fi Cauchy.
fi fi fi fi
bn
bn
+1
0 n .
3.43. Y fi:
()|c1| |c2| |c3| .() c2m1 0, c2m 0 (m= 1, 2, 3, . . .).() limncn= 0.Tfi,
n=1cn .
O () fi
. T fi fi Leibniz2.
2 . . M.: Gottfried Wilhelm von Leibniz (1646-1716). fi . Efi
M, , ,
, N, , I, E, .
fi .
A. M Isaac Newton (1642-1727)
A . E,
A. E fi fi
fi M. Mfi
fi fi Leibniz fi
Alfred North Whitehead (1861-1947) Bertrand Russell (1872-1970),
George Boole (1815-1864).
fi Leibniz E, -. E 1661 fi
N, fi fi .
1663. T fi , fi N
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110
Afi. Efi 3.42 an = (1)n+1, bn = |cn|
(n= 1, 2, 3, . . . ).
3.44. Y fi
n=0cnzn 1 fi c0 c1 c2 , limncn = 0. Tfi,
n=0cnzn z |z| =1, fi fi z= 1.
Afi. an= zn, bn= cn (n= 1, 2, 3, . . . ). O 3.42 , fi
|An| = n
m=0zm
= 1 zn
+1
1 z 2|1 z| (n= 1, 2, 3, . . . )
|z| = 1 z= 1.
AOYTH YKIH
H
n=1an fi fi
n=1 |an| . . T 1666, . O Leibniz , , fi
Altdorf. E, N, .
O Leibniz
Mainz 1672. T fi fi fi
1676. , Leibniz M
fi Christiaan Huygens (1629-1695). T 1673,
Leibniz B E .
O Leibniz 1676 fi
Braunschweich-Lneberg. , ,
Braunschweich-Lneberg. E fi Braunschweich-Lneberg
.O Leibniz A E B 1700
fifi . T Leibniz A E
.
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AKOOYIE KAI EIPE 111
3.45. E
n=1an , fi
n=1an
.
Afi. T fi fim
k=nak
m
k=n|ak| (m n)
Cauchy.
3.46. fi, fi
.
fi n=1an fi n=1an n=1 |an| . E ,
n=1
(1)nn
( 3.43).
T , fi fi -
fi
. H -
. I,
fi .
, fi -
fi .
M fi fi
. , ,
fi fi
fi .
POEH KAI OAAIAMO EIPN
3.47. E
n=1an= A
n=1bn= B, fi
n=1(an+bn) = A + B
n=1can= c A, fi fi fi c.
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112
Afi. A
An=n
k=1ak, Bn=
nk=1
bk (n= 1, 2, 3, . . . ) .
Tfi,
An+Bn=n
k=1(ak+ bk) (n= 1, 2, 3, . . . ) .
Efi limn An= A limnBn= B, fi
limn
(An+Bn ) = A + B .
H fi fi .
, fi
fi
. H fi fi
. A, fi. Afi
fi. fi Cauchy fi.
Ofi 3.48.
n=0an
n=0bn ,
cn=n
k=0
akbnk (n= 0, 1, 2, . . . )
n=0cn fi .
O fi fi : -
n=0anzn
n=0bnz
n , fi fi
fi z , fi
n=0
anzn
n=0
bnzn = (a0 + a1z + a2z2 + )(b0 + b1z + b2z2 + )
= a0b0
+(a0b1
+a1b0)z
+(a0b2
+a1b1
+a2b0)z
2
+ = c0 + c1z + c2z2 + .
z= 1, fi.
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AKOOYIE KAI EIPE 113
3.49. E
An=n
k=0ak, Bn=
nk=0
bk, Cn=n
k=0ck (n= 0, 1, 2, . . . )
An A, Bn B n , fi fi {Cn} (n=0, 1, 2, . . . ) A B, fi (-) fi Cn= AnBn fi fi n. H {Cn}(n= 0, 1, 2, . . . ) fi {An} (n= 0, 1, 2, . . . ) {Bn} (n= 0, 1, 2, . . . ) ( fi 3.50).
fi fi .
H
n=0
(1)nn + 1
= 1 12+ 1
3 1
4+
( 3.43). fi
fi
n=0
cn = 1
12+ 1
2
+
1
3+ 1
2
2+ 1
3
14+ 1
3
2+ 1
2
3+ 1
4
+
,
cn= (1)nn
k=0
1(n k+ 1)(k+ 1) (n= 0, 1, 2, . . . ) .
Efi
(n k+ 1)(k+ 1) =n
2+ 1
2
n2 k
2
n2+ 1
2,
fi
|cn
|
n
k=0
2
n + 2=
2(n + 1)
n + 2 (n
=0, 1, 2, . . . ) ,
cn0 n ,
n=0cn , .
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114
fi ,
Mertens3
, fi - .
3.50. Y fi:
() H
n=0an .
()
n=0an= A.()
n=0bn= B.
() cn=n
k=0akbnk (n= 0, 1, 2, . . .).Tfi,
n=0 cn= A B.
, fi ,
, fi .
Afi.
An=n
k=0ak, Bn=
nk=0
bk, Cn=n
k=0ck, n= Bn B (n= 0, 1, 2, . . . ) .
Tfi, n= 0, 1, 2, . . . fiCn = a0b0 + (a0b1 + a1b0) + + (a0bn+ a1bn1 + + anb0)
= a0Bn+ a1Bn1 + + anB0= a0(B+ n ) + a1(B+ n1) + + an (B+ 0)= AnB+ a0n+ a1n1 + + an 0.
n
=a0n
+a1n
1
+ +an0 (n
=0, 1, 2, . . . ) .
3 . . M.: Franz Mertens (1840-1927). fi
A . E Leopold Kronecker (1823-1891)
Ernst Kummer (1810-1893).
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AKOOYIE KAI EIPE 115
E fi Cn A Bn . Efi AnB A B
n , filim
nn= 0. (21)
=
n=0|an|.
(E ().) A >0.
(), n 0 n . , fi N n N, fi|n| . , n N
|n| |0an+ + NanN| + |N+1anN1 + + n a0| |0an+ + NanN| + .
Nfi n fi
lim supn
|n| ,
fi ak 0 k . Efi fi fi, (21).
n=0cn
A B fi . O Abel4 fi
.4 . . M.: Niels Henrik Abel (1802-1829). Nfi fi,
fi .
Kfi fi , Abel fi fi
, fi o ,
fi .
T 1821 Abel X (fi Oslo), fi fi
1822. fifi
fi . , , Bernt
Michael Holmbo (1795-1850), Abel fifi .
1824, fi
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116
3.51. E
n=0an ,
n=0bn ,
n=0cn
A, B, C cn= a0bn + +an b0 (n= 0, 1, 2, . . .),fi C= A B.
fi fi
. fi (
) fi 8.2.
ANAIATAEI
fi Carl Friedrich Gauss (1777-
1855). fi
, fi fi fi
, fi Gauss, ,
fi Abel. O Gauss, fi
fi ,
Abel, . , Abel 1826
A E .
Adrien Marie Legendre (1752-1833)
Augustin Louis Cauchy (1789-1857). O Abel. ,
fi fi . Z
fi Abel , fi .
fi , Abel 1827 X
fi fi
. T X
fi .
1828 Abel Florand, , fi
.
Abel, o August Leopold Crelle (1780-1856),
Journal fr die Reine und Angewandte Mathematik,
M ,
fi B
M.
Charles Hermite (1822-1901), Abel fi -
fi .
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AKOOYIE KAI EIPE 117
Ofi 3.52. A {kn} (n = 1, 2, 3, . . . )
, fi fi fi ( {kn} 1-1 fi J J, Ofi 2.2).
an= akn (n= 1, 2, 3, . . . ) ,
n=1an
n=1an.
E{sn}, {sn} (n = 1, 2, 3, . . . )
n=1an
n=1a
n, fi
fi, , fi
. , fi fi
.
3.53.
1 12+ 1
3 1
4+ 1
5 1
6+ (22)
1 + 13 1
2+ 1
5+ 1
7 1
4+ 1
9+ 1
11 1
6+ , (23)
fi fi fi fi. E s (22), fi
s 0
k1, fi s 3
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118
T fi fi fi
, Riemann5
.
3.54.
n=1an
5 . . M: Georg Friedrich Bernhard Riemann (1826-1866). fi
fi. E A, M . Afi
Riemann Albert Einstein (1879-1955)
fi. O Riemann fi fi
fi , .
O Riemann fi fi . O
. Afi fi
M, .
, X, Riemann Gttingen 1846 fi
. fi , Riemann
M, . B,
fi fi fi Gustav Peter Lejeune Dirichlet (1805-1859),
Jacob Steiner (1796-1863), Carl Gustav Jacob Jacobi (1804-1851) Ferdinand Gotthold
Max Eisenstein (1823-1852). O Riemann Gttingen 1849
. T 1851
Riemann Carl Friedrich Gauss (1777-
1855), , fi fi. T , Riemann
Gttingen, fi fi
Gauss.
Gauss 1855, Dirichlet
Gttingen Riemann. H ,
, 1856 Riemann . T
fi Riemann Gttingen fi
Dirichlet , fi
.
T 1862 Riemann fi fi .
, I. E fi I
. T Pisa,
. E ,
I 1863, fi fi . fi
I , I, fi 1866. T Riemann, I
: fi
fi.
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AKOOYIE KAI EIPE 119
. Y fi
+.
Tfi,
n=1an -
sn (n= 1, 2, 3, . . .)
liminfn
sn= , limsupn
s n= . (24)
Afi.
pn=|an| + an
2 , qn=
|an| an2
(n= 1, 2, 3, . . . ) .
Tfi, pn qn= an, pn+ qn= |an|, pn 0, qn 0, n. O
n=1 pn
n=1qn .
E , fi
n=1
(pn+ qn ) =
n=1|an|
, fi fi. Efi
N
n=1an=
N
n=1(pn qn ) =
N
n=1pn
N
n=1qn (N= 1, 2, 3, . . . ) ,
fi
n=1 pn
n=1qn ( fi) -
fi
n=1an , fi fi.
A P1,P2,P3, . . . fi
n=1an ,
, Q1,Q 2,Q 3, . . . fi
fi
n=1an , .O
n=1 Pn,
n=1 Q n fi
n=1 pn ,
n=1qnfi fi .
{mn},{kn}(n
=1, 2, 3, . . . )
P1 + +Pm1Q 1 Q k1++Pm1+1 + +Pm2Q k1+1 Q k2+ ,
(25)
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AKOOYIE KAI EIPE 121
fip 1, 2, . . . ,N
k1, k2, . . . , kp ( fi O- 3.52). Tfi, n > p, a1, . . . , aN
sn sn |sn sn| , fi (26). , {sn}(n= 1, 2, 3, . . . ) {sn} (n= 1, 2, 3, . . . ) n .
AKHEI
1. A fi {sn} (n=1, 2, 3, . . . )
{|sn|} (n= 1, 2, 3, . . . ). A ;
2. Y limn(
n2 + n n).
3. Es1=
2
sn+1=
2 + sn (n= 1, 2, 3, . . . ) ,
fi fi {sn} (n= 1, 2, 3, . . . ) fi sn < 2 n= 1, 2, 3, . . . .
4. Y fi {sn}(n= 1, 2, 3, . . . ), fi
s1= 0, s2m=s2m1
2 , s2m+1=
1
2+ s2m (m= 1, 2, 3, . . . ) .
5. {an},{bn} (n= 1, 2, 3, . . . ) fi
limsupn
(an+
bn
)
lim supn
an+
limsupn
bn
,
fi .
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122
6. fi
n=1an,
()an=
n + 1 n,()an=
n + 1 n
n ,
()an= ( n
n 1)n ,()an= 11 + zn , fiz fi fi
.
7. A fi
n=1an
n=1
an
n
fian 0 n .
8. E
n=1an {bn} (n = 1, 2, 3, . . . ) , fi
n=1an bn .
9. Y fi
:
() n=0n3zn .()
n=0
2nn!z
n .
()
n=12n
n2zn .
()
n=0n3
3nzn .
10. Y fi
n=0anz
n
, fi
fi 0. A fi
1.
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AKOOYIE KAI EIPE 123
11. Y fi an >0 sn= a1+ . . . + an
n fi n=1an.
() A fi
n=1an
1 + an .() A fi
aN+1sN+1
+ + aN+ksN+k
1 sNsN+k
N k
fi
n=1ansn
.
() A fian
s2n 1
sn
1 1
sn
n fi
n=1ans2n
.
() T
n=1
an
1 + nan
n=1
an
1 + n2an;
12. Y fi an > 0 (n= 1, 2, 3, . . . ) fi
n=1an.
rn= m=n
am (n= 1, 2, 3, . . . ) .
() A fi
am
rm+ + an
rn>1 rn
rm,
m, n m < n fi
n=1anrn
.
() A fian
rn
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124
13. A fi Cauchy fi -
.
14. E {sn} (n= 0, 1, 2, . . . ) fi,fi fi ,n(n= 0, 1, 2, . . . ), fi
n=s0 + s1 + + sn
n + 1 , (n= 0, 1, 2, . . . ) .
() E limnsn= s, fi fi limn n= s.() K {sn} (n=0, 1, 2, . . . )
limn n= 0.
() E fi sn >0 n limsupnsn= , fi fi limn n= 0;
() an= sn sn1 n= 1, 2, 3, . . . . fi
sn n=1
n + 1n
k=1kak
n. Y fi limn(nan ) = 0 fi {n}(n=0, 1, 2, . . . ) . A fi {sn}(n=0, 1, 2, . . . ) .(Afi (), fi fi fi
nan
0 n
.)
() A -
fi: Y fi M fi fi
|nan| M n, fi limn n = . , fi limnsn= , fifi :
Em
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AKOOYIE KAI EIPE 125
> 0 fi n
fim
m n 1 +
x2,x3,x4, . . .
xn+
1
=
1
2xn+
xn (n= 1, 2, 3, . . . ) .
() A fi {xn} (n = 1, 2, 3, . . . ) filimnxn=
.
() n= xn
(n= 1, 2, 3, . . . ). A fi
n+1=2n
2xn x5 > .() A fi x2
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AKOOYIE KAI EIPE 127
21. A fi 3.10(): E
{En} (n= 1, 2, 3, . . . ) fi fi X, En+1 En n
limn
diamEn= 0,
fi
n=1En fi .
22. Y fi X fi fi
{Gn} (n= 1, 2, 3, . . . ) -fi X. A Baire fi
n=1G n
fi. ( fi, fi X.)Yfi:B {En} (n=
1, 2, 3, . . . ) En Gn n fi (21).
23. Y fi{pn},{qn} (n= 1, 2, 3, . . . ) Cauchy fi X. fi {d(pn , qn )} (n=1, 2, 3, . . . ) .
Yfi: n , m fi
d(pn, qn ) d(pn ,pm ) + d(pm , qm) + d(qm , qn ).
Afi fi fi fi
|d(pn, qn ) d(pm , qm )|
, fi n, m .
24. A X fi .
() O Cauchy
{pn
},
{qn
}(n
=1, 2, 3, . . . )
X fi
limn
d(pn , qn ) = 0.
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128
A fi .
() A X
6
. E P X, Q X {pn} P , {qn} Q, fi
(P,Q) = limn
d(pn , qn ).
23, fi . A fi fi
(P,Q) {pn},{qn}(n= 1, 2, 3, . . . ) - fi fi
X.
() A fi fi X .
() p X Cauchy, fi fi p. Y fi Pp X
. A fi
(Pp,Pq ) = d(p, q)
p, q X. M , fi (p)= Pp (p X) ( fi ) fi X X.
() A fi (X) fi X fi (X)= X
X . (), (X) X Xfi X.
O X X.
25. A X fi ,
d(x,y )= |x y|(x,y X). ;( 24.)
6 . . M.: A Ex E. T Cxfi y E y x x( ).
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K 4
YNEXEIA
H fi
O 2.1 2.2. A fi
(
-
), (
Rk) fi
fi fi . T
fi fi , -
, . Afi, fi fi fi
fi .
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T
, .
OPIA YNAPTHEN
Ofi 4.1. X, Y. Y fi EX, fi f E Y fi p fi
E. f(x) qx p
limxp
f(x) = q (1)
fi q Y fi fi: >0 >0
dY(f(x), q) < (2)
x E
0< dX(x,p) < . (3)
(E, dX, dY X, Y.)
, q fi f p.
E X / Y , fi
E Rk, fi dX, dY
( Efi 2.16).
fi fi p X, p E. E, fi
p E, f(p) = limxp f(x).
M fi
:
4.2. Y fi X, Y,E, f p fi
fi Ofi 4.1. Tfi,
limxp
f(x) = q (4)
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YNEXEIA 131
fi
limn
f(pn ) = q (5)
{pn} (n= 1, 2, 3, . . .) E
pn= p (n= 1, 2, 3, . . . ) , limn
pn= p. (6)
Afi. Y fi (4). {pn} (n=1, 2, 3, . . . ) E, (6). A >0. Tfi,
>0 dY(f(x), q) < x E 0 < dX(x,p) < . E, fiN n > N, fi 0< dX(pn ,p ) < .
, n n > N fi dY(f(pn), q) < , fi
fi (5).
Afi, fi (4) . Tfi,
> 0 > 0 x E ( o fi ), fi dY(f(x), q) 0 < dX(x,p) < . n= 1/n (n= 1, 2, 3, . . . ), E (6) fi (5).
fi. E f fi p, fi fi fi.
Afi fi 3.2() 4.2.
Ofi 4.3. Y fi f, g, - E. f+ g x E fi f(x) + g(x). M fi fi f g, fi f g f/g ,fi fi fi fi x E
g(x)=0. H f fi x E fi c. f= c. E f, g ,fi f g fi f(x) g (x) x E.
, f
,g
E Rk
, fi f + g f g
(f + g)(x) = f(x) + g(x), (f g)(x) = f(x) g(x) (x E).
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E fi fi, fi f
(f)(x) = f(x)(x E).
4.4. Y fi X fi , fi EX, fi p fi E, fi f, g
E fi
limxp
f(x) = A, limxp
g(x) = B.
Tfi:
() limx
p(f
+g)(x)
= A
+B.
() limxp(f g)(x) = A B.() limxp
fg
(x) = A
B B= 0.
Afi. B 4.2, fi
fi ( 3.3).
.E f, g E Rk, fi ()
()
(') limxp(f g)(x) = A B.( 3.4.)
YNEXEI YNAPTHEI
Ofi 4.5. A X, Y , E X, p E f EY. H f p
fi >0 >0
dY(f(x), f(p)) <
x EdX(x,p) < .H f E fi
E.
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134
Afi. A > 0. Efi g f(p),
>0 dZ(g(y), g(f(p))) < y f(E) dY(y, f(p)) < .
Efi f p, >0
dY(f(x), f(p)) < x E dX(x,p) < .
Afi fi
dZ(h(x), h(p)) = dZ(g(f(x)), g(f(p))) <
x
E dX(x,p) < . , .
4.8. M fi f fi X
fi Y X fi f1(V)
fi X fi V Y.
(O fi Ofi 2.2.) T
fi .
Afi. Y fi f X fi V fi
Y. fi f
1(V)
fi f1(V). , p X f(p)V. Efi V fi, > 0 y V dY(f(p),y) < . Efi f p, > 0
dY(f(x), f(p) ) < dX(x,p ) < . , x f1(V) dX(x,p) < .
Afi, fi f1(V) fi
X fi V Y. p X > 0. A V y Y dY(y, f(p)) < . Tfi, V fi., f1(V) fi. K , >0
x f1(V) dX(p,x) < . , x f1(V), fi f(x) V dY(f(x), f(p)) < .
Afi fi.
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YNEXEIA 135
fi. M fi f fi X fi
Y X fi f1
(C) fi X, fi C Y.
Afi fi , fi
fi fi fi fi
f1(Ec) = [f1(E)]c EY.Efi
Rk.
4.9. A f, g
fi X. Tfi, f+ g, f g f/g X.
, fi g(x)=0, x X.
Afi. o fi fi -
. fi fi 4.4
4.6.
4.10.
() A f1, . . . , fk ,
fi X. Y fi f fi X Rk
fi fi
f(x) = (f1(x), . . . , fk(x)) (x X). (7)
Tfi, f fi fi -
f1, . . . , fk .
() E f, g X Rk, fi
f + g, f g X.
O f1, . . . , fk f. - fi f + g R k, f g .
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136
Afi. T () fi fi
|fj (x) fj (y)| |f(x) f(y)| =
ki=1
|fi (x) fi (y)|2
12
j= 1, . . . , k x X. T () fi () 4.9.
4.11. E x1, . . . ,xk fi x
Rk, fi i (i= 1, . . . k) fi
i (x) =xi (i= 1, . . . k, x Rk) (8)
Rk, fi fi
|i (x) i (y)| |x y| (i= 1, . . . k, x, y Rk)
fi = Ofi 4.5. O i (i= 1, . . . k) .
M 4.9 fi
xn11 x
n22 xnkk , (9)
fi n1, . . . , nk , -
Rk. T (9),
fi . E,
P , fi
P(x) =
cn1nkxn11 xnkk (x Rk), (10)
R k. E, cn1nk
, n1, . . . , nk (10)
fi.
E, x1, . . . ,xk, (10), R k,
fi 0.
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YNEXEIA 137
Afi fi fi
||x| |y|| |x y| (x, y Rk). (11)
, fix |x|(x Rk) - Rk.
E f fi fi fi X
Rk, fi fi fi fi (p) = |f(p)| (p X), fi 4.7, X.
4.12. H f
Efi X.
, X fi fi fi
( fi fi ).
E,
f. Afi fi
fi ,
. Afi
fi .
4.8 4.10
fi fi.
YNEXEIA KAI YMAEIA
Ofi 4.13. M fi f fi fi E Rk
fi fi fi
|f(x)| M x E.
4.14. Y fi f fi fi
X fi Y. Tfi, f(X)
.
Afi. A {Va}(a A) f(X). Efi f , 4.8 fi fi
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138
f1(Va) (a A) fi. Efi X ,
1, . . . , n
X f1(V1 ) f1(Vn ). (12)
Efi f(f1(E)) E E Y, (12) fi
f(X) V1 Vn . (13)
Afi fi.
: f(f1(E)) E,
E Y. E E X, fi E f1
(f(E)). fi.
E , 4.14.
4.15. E f fi fi
X Rk, fi f(X) fi .
, f .
To fi 2.41. T fi
fi fi f :
4.16. Y fi f -
, fi X.
M= suppX
f(p), m= infpX
f(p). (14)
Tfi, p, q X f(p) = M f(q) = m.
O fi (14) fi M
fi f(p), fi p X, m fi.
T : - p, q f(q) f(x) f(p) x X. , f ( p) ( q ).
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YNEXEIA 139
Afi. 4.15, f(X) fi -
. , f(X)
M= sup f(X) m= inf f(X),
2.28.
4.17. Y fi f 1-1 fi
fi X fi Y. Tfi,
fi f1, Y fi
f
1(f(x))=
x (x
X),
fi Y X.
Afi. Efi 4.8 f1
fi fi f(V) fi
Y fi V X. fi fi
V X.
T Vc V fi, ( 2.35).
K , f(Vc) Y( 4.14)
fi. Efi f 1-1, f(V)
f(Vc). , f(V) fi.
Ofi 4.18. Y fi f fi fi
X fi Y. H f fi X
fi >0 >0
dY(f(p), f(q)) < (15)
p, q XdX(p, q) < .
A fi . E , fi fi
,
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140
. Ofi . Kfi,
f X, fi >0 p X >0 fi Ofi 4.5.
, fi p. E fi f fi
, fi > 0 fi > 0
fi X.
K fi, fi
. T fi
.
4.19. Y fi f fi fi
X fi Y. Tfi, f fi .
Afi. >0. Efi f ,
p X fi fi (p)
q X, dX(p, q) < (p), fi dY(f(p), f(q)) 0, f fi
E.
H f fi
g(x) = 11 + (x x0)2
(x E) (22)
E , fi 0 < g(x) < 1 x E.E fi
supxE g(x) = 1,
g (x)
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YNEXEIA 143
1, fi fi
f(t) = (cos t, sin t) (0 t
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144
, fi f(a) > f(b). M fi
, fi fi .
Afi. 2.47, [a, b] fi. ,
4.22 fi f([a, b]) fi
R1 fi
2.47.
4.24. E fi, fi 4.23
. x1,x2 x1 < x2
fi fi c f(x1), f(x2) x
(x1,x2) f(x) = c, fi f . T 4.27() fi fi .
EIH AYNEXEIN
Ex f, fi f
x fi x fi
f x. E f , fi
.
fi , fi f x,
f(x+) f(x).
Ofi 4.25. Y fi f (a, b). -
xa x
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YNEXEIA 145
f x f(x) a < x b, fi {tn}
(n= 1, 2, 3, . . . ) (a,x).E fi x (a, b) :
limtx f(t) fi
f(x+) = f(x) =limtx
f(t).
Ofi 4.26. Y fi f (a, b). H f
fi x
x f(x+), f(x). ( fi fi f(x+), f(x)
), .Y f -
: f(x+)= f(x) (fi f(x) fi) f(x+) = f(x) = f(x).
4.27.
() O f
f(x) =
1 x fi fi,
0 x fi.
Tfi, f xfi
f(x+) f(x).() O f
f(x) =
x x fi fi,
0 x fi.
Tfi, f x=0 .
() O f
f(x) =
x+ 2 3< x < 2,x 2 2 x
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146
Tfi f x= 0
(3, 1).() O f
f(x) =
sin1x x= 0,0 x= 0.
Efi f(0+) f(0) , f x= 0. fi fi sin . E fi , fi 4.7
fi f x x= 0.
MONOTONE YNAPTHEI
.
Ofi 4.28. A f (a, b). H f
(a, b) fi
x,y a < x < y < b fi f(x) f(y). H f
(a, b) fi fi . M fi fi
.
4.29. A f (a, b). Tfi,
f(x+), f(x) x (a, b). , fi
supa
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YNEXEIA 147
A, -
.
Afi. fi , f(t)
a < t < x fi fi f(x)
, A. ,
A f(x). fi A= f(x). > 0. Afi fi A >0
a < x < x
A < f(x ) A. (27)
Efi f , fi
f(x ) f(t) A (x
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148
T fi fi fi fi -
. A fi , fi
17, fi fi fi
.
4.30. A f (a, b). Tfi,
f
.
Afi. Y fi f ( fi
fi ). A E
f. x E fi fi r(x)
f(x)
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YNEXEIA 149
, :
n fi xn < x. E , fi 0. Efi (31) ,
fi .
O :
() H f (a, b).
() H f E. , fi
f(xn+) f(xn) = cn (n= 1, 2, 3 . . . ) .
() H f (a, b).
E, fi f(x
)=
f(x)
x (a, b). M fi
. E (31) n
xn x, fi fi f(x+)= f(x) x (a, b). , .
fi
fi. 6.16.
AEIPA OPIA KAI OPIA TO AEIPO
-
, Ofi 4.1
.
fi fi x x
(x ,x+ ), fi >0.
Ofi 4.32. T x x > c, fi c
fi fi, +. fi (c,
+). ,
x x < c fi
(, c).
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150
Ofi 4.33. A f ,
E.
f(t) A t x,
fiA x ,
fi U A V x
V E fi f(t)U t V Et= x. , fi Afi f x.
fi fi Ofi
4.1 fi A x .
T 4.4 , fi
. fi, :
4.34. Y fi f, g ,
E. Y fi
f(t) A, g(t) B t x,
fi A,B x
. Tfi:
() E f(t) A t x, fi A , fi A= A.
() (f+ g)(t) A +B t x.() (f g)(t) A B t x.() (f/g)(t) A/B t x.B, fi (), ()
() .
fi , 0, / 0/0 ( Ofi 1.23).
AKHEI
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YNEXEIA 151
1. Y fi f ,
R1
, fi
limh0
[f(x+ h) f(x h)] = 0
x R1. E f ;
2. E f fi fi X
fi Y, fi fi
f(E) f(E)
E X. (M E E.)
fi f(E)
f(E).
3. Y fi f
fi X. Z(f)(
f) p X f(p)= 0.A fi Z(f) fi.
4. Y fi f, g fi
X fi Y fi E fi
X. A fi f(E) fi f(X). Eg (p)= f(p) p E, fi fi g(p)= f(p) p X. (M , fi fi
fi .)
5. E f ,
fi E R1, fi fi
g, R1, g(x)= f(x) x E.(M fi f
fi E R 1.) fi
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152
fi. E fi
.Yfi: fi 1 g
fi E
( 29 K 2). T
R1 fi fi , fi
fi fi .
6. E f E, fi
f (x, f(x)) x E. I, E f , fi
.Y fi E fi f
, E. E E ,
fi fi f fi
.
7. EE X f X, fifi f E g , E,
fi g(p) = f(p) p E.
O f, g R2
: f(0, 0)=g(0, 0) = 0 f(x,y) = x y2/(x2+y4), g(x,y) = x y2/(x2+y6) (x,y ) = 0.A fi f R2, fi g
(0, 0) fi f (0, 0).
' fi , f,g R2
!
8. Y fi f fi
, E R1. A fi
f E.
fi fifi E .
1 . . M.: fi fi .
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YNEXEIA 153
9. fi fi fi
: >0 >0 diamf(E) < E X diamE < .
10. fi
4.19: E f fi ,
fi > 0 {pn},{qn} (n= 1, 2, 3, . . . ) dX(pn , qn ) 0 n dY(f(pn ), f(qn)) > n. X 2.37
.
11. Y fi f fi fifi X fi Y. A fi {f(xn )}(n=1, 2, 3, . . . ) Cauchy Y Cauchy{xn} (n= 1, 2, 3, . . . ) X. X fi fi
13.
12. M fi fi
fi .
.
13. A E fi fi X
f fi ,
E. A fi f fi E X
( 5 ). (H fi
fi 4.)
Yfi: p X fi fi n, Vn (p) q E d(p, q) < 1/n. X 9 fi fi
f(V1(p)), f(V2(p) ) , . . . fi fi g(p)
R1. A fi g f X.M R1 fi Rk; Afi fi
; Afi fi ;
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154
14. A I= [0, 1]. Y fi f
I I. A fi ( ) x I f(x) = x.
15. O fi ffi X
fi Y fi f(V) fi
fi V X.
A fi fi R1 R1
fi.
16. fi fi x [x] - fi fi x, [x]
fi fi x 1 < [x] x. A (x)=x [x], fi x. [x] (x);
17. Y fi f ,
(a, b). A fi fi f
.
Yfi:A E x f(x) < f(x+). x E (p, q, r)
fi:
() f(x) < p < f(x+).() Ea
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YNEXEIA 155
n >0. x= 0, fi n=1. f
R1
fi fi
f(x) =
0 x fi,
1/n x= m/n, fi .A fi f fi fi f
fi fi.
19. Y fi f ,
R1, fi : a, b, c
f(a) r > f(x0) fi fi r n, fi n tn
x0 xn f(tn)= r. ,tn x0n . K . (N. J. Fine,Amer. Math. Monthly, vol. 73, 1966, p.782.)
20. E E fi fi X,fi fi x Xfi E fi
E(x) = infzE
d(x,z).
() A fiE(x) = 0 fi x E.() A fi E fi X
fi
|E(x) E(y)| d(x,y) x,y X.
Yfi: x,y,z X fi E(x)d(x,z)d(x,y ) + d(y,z)
E(x) d(x,y ) + E(y).
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156
21. Y fi K F
fi X, fi K fi F fi.A fi >0 d(p, q) > p K,q F.
Yfi:H F K.
fi
.
22. Y fi A, B -
fi X. O f
f(p)
=A(p)
A(p) + B (p)(p
X).
A fi f X fi-
fi [0, 1], fi f(p) = 0 fi p A fi f(p) = 1 fi p B.
T A 13: K fi
A X Z(f)
f X.
V= f1([0, 1/2)), W= f1((1/2, 1]),
fi V,W fi AV,B W. (, fi , fi
fi. H fi fi2 .)
2 . . M.: O fi fi normality. ,
T, fi
fi regular normal
fi, fi , fi. normality ,
fi , fi fi. -
, fi fi fi fi , fi fi .
, fi normal vector, fi normal fi. K
T normal spaces.
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YNEXEIA 157
23. M f, (a, b),
fi
f(x+ (1 )y) f(x) + (1 )f(y)
x,y(a, b) fi fi 0<
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158
26. Y fi X,Y, Z fi Y
. A f fi X Y g 1-1fi Y Z. fi h h(x)= g(f(x)) x X.
A fi f fi h fi
.
Yfi: H g1 g(Y) f(x) =g1(h(x)).
A fi f h .
( 4.21
fi ) fi Y -
fi fi, fi X Z .
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K 5
IAOPIH
fi (fi fi fi)
, .
Afi fi , ' fi -
fi fi -
. H fi E
K 9.
H APAO PAMATIKH YNAPTHE
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160
Ofi 5.1. A f , [a, b], .
x [a, b] (t) = f(t) f(x)
t x (a
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IAOPIH 161
Afi. 4.4, t x fi
f(t) f(x) = f(t) f(x)t x (t x) f
(x) 0 = 0.
T . E
-
. K 7
o R1 !
5.3. Y fi f, g -
[a, b] fi x [a, b]. Tfi, f+ g, f g, f/g x :
() (f+ g)(x) = f(x) + g(x).
() (f g)(x) = f(x)g(x) + f(x)g(x)
()
fg
(x) = g(x)f
(x) g(x)f(x)g2(x)
. E
fi g(x) = 0.
Afi. T () , 4.4.
A h= f g. Tfi, t [a, b] t= x,
h(t) h(x) = f(t)[g(t) g(x)] + g(x)[f(t) f(x)].
E fi t x fif(t) f(x) t x ( 5.2), fi (). E, h= f/g. Tfi, t [a, b] t= x,
h(t) h(x)
t x =
1
g(t)g(x)g(x) f(t) f(x)
t x f(x)
g(t) g(x)
t x . t x fi 4.4 5.2, ().
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162
5.4. H -
0. E f f(x)= x x , fi f(x)=1. M - () () 5.3 fi xn
nxn1 fi fi
n ( n < 0, fi x= 0). , fi , fi fi
.
T fi fi fi
. E fi
fi . K 9 fi .
5.5. Y fi f [a, b],
fi x [a, b] f(x), fi g I f
fi f(x). E
h(t) = g (f(t)) (a t b),
fi h x fi
h(x) = g (f(x))f(x). (3)Afi. A y= f(x). fi , u ,v
f(t) f(x) = (t x)[f(x) + u(t)], (4)
g(s) g(y) = (sy )[g(y) + v(s)], (5)
fit [a, b],s Iu (t) 0 t x,v(s) 0 s y. As= f(t). X (5) (4), fi
h(t) h(x) = g(f(t)) g(f(x))= [f(t) f(x)] [g(y) + v(s)]= (t x) [f(x) + u(t)] [g(y) + v(s)],
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IAOPIH 163
, fit= x,
h(t) h(x)t x = [g
(y) + v(s)] [f(x) + u(t)]. (6)
t x, fi s y, fi f, (6) g(y)f(x),
(3).
5.6.
() O f
f(x) = xsin1x x= 0,0 x= 0. (7)
fi fi sin cos (
K 8),
fi 5.3 5.5 x= 0 fi
f(x) = sin1x
1x
cos1
x(x= 0). (8)
x=0 fi fi 1/x . K fi: t
=0
f(t) f(0)t 0 = sin
1
t.
T fi, t 0, f 0.
() O f
f(x) =
x2 sin1x x= 0,0 x= 0.
(9)
, fi
f(x) = 2xsin1x
cos1x
(x= 0). (10)
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x= 0, fi fi f(t) f(0)t 0
=tsin1t
|t| (t= 0).t 0, fi
f(0) = 0. (11)
, f x, f
fi fi cos(1/x) (10) x 0.
EPHMATA MEH TIMH
Ofi 5.7. A f ,
fi X. H f fi fi p X fi >0 f(q) f(p), q X d(p, q) < .
T fi.
T fi -
.
5.8. A f ,
[a, b]. E f fi x (a, b) f(x), fi f(x) = 0.
, fi .
Afi. E >0 fi Ofi 5.7
a < x < x< x+
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IAOPIH 165
Ex h (a) t (a, b), fi x [a, b] h ( 4.16). (12), x (a, b) 5.8 fi h(x)= 0. E h(t) < h(a) t (a, b), fi fi x [a, b] h .
T . H fi
:
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166
5.10. A f , -
[a, b] (a, b). Tfi, x (a, b)
f(b) f(a) = (b a)f(x).
Afi. g (x) = x(x [a, b]) 5.9.
5.11. Y fi f ,
(a, b).
() E f(x) 0 x (a, b), fi f .() E f(x) = 0 x (a, b), fi f .() E f(x) 0 x (a, b), fi f .
Afi. H fi -
fi
f(x2) f(x1) = (x2 x1)f(x),
x1,x2 (a, b)x x1,x2.
YNEXEIA TN APAN
( 5.6()) fi f
f
, . ,
. I,