realsum14p

!

http://eclass.uoa.gr/courses/MATH244/

2014-15

X . X : X X R :(i) (x ,y) 0 x ,y X (x ,y) = 0

x = y .

(ii) (x ,y) = (y ,x) x ,y X .(iii) (x ,z) (x ,y) + (y ,z) x ,y ,z X (

).

(X ,) .

() R

d(x ,y) = |xy |, x ,y R.

() Rm, m- x = (x1, . . . ,xm) , :

2(x ,y) =

(n

i=1

(xi yi )2)1/2

.

() X : : X X R

(x ,y) ={

1, x 6= y0, x = y

. .

() X : f : X R 1-1, df X :

df (x ,y) = |f (x) f (y)|, x ,y X .

() n- Hamming.

Hn = {,}n ={

(x1,x2, . . . ,xn) xi = , i = 1, . . . ,n}.

h : HnHn R, h(x ,y) n- x = (x1, . . . ,xn) y = (y1, . . . ,yn),

h(x ,y) = card({

1 i n : xi 6= yi})

.

(Hn,h

) Hamming.

( )

(X ,) . A X , A : AA R

A(x ,y) = (x ,y), x ,y A

( AA) A, A. (A,A) .

, R .

()

(X ,) . A X C > 0 x ,y A (x ,y) C .,

sup{(x ,y) : x ,y A}< .

, A

diam(X ) := sup{(x ,y) : x ,y A}.

diam( /0) = 0. {(x ,y) : x ,y A} sup{(x ,y) : x ,y A}= + diam(A) = +.

:

() R d(x ,y) = |xy | .

() R

(x ,y) =|xy |

1 + |xy |, x ,y R

, (x ,y) < 1 x ,y R. diam(R,) = 1.

() X , ..(X , ) (, X = {} diam(X ) = 0, diam(X ) = 1).

(X ,d) ..

(x ,y) =d(x ,y)

1 + d(x ,y), x ,y X

X .. . (x ,y) = f (d(x ,y)) f (t) = t1+t , t 0. f , , d(x ,y) d(x ,z) + d(z ,y) f (d(x ,y)) f (d(x ,z) + d(z ,y)),

(x ,y) d(x ,z) + d(z ,y)1 + d(x ,z) + d(z ,y)

=d(x ,z)

1 + d(x ,z) + d(z ,y)+

d(z ,y)

1 + d(x ,z) + d(z ,y)

d(x ,z)1 + d(x ,z)

+d(z ,y)

1 + d(z ,y)= (x ,z) + (z ,y).

()

X (:) . X : X R :

() x 0 x X x= 0 x = 0( ).

() x= | | x R x X ().

() x + y x+y x ,y X ().

X , (X , ) . d : X X R

d(x ,y) = xy, x ,y X

( X ).

1. Rm supremum : Rm R : x = (x1, . . . ,xm) Rm

x := max{|xi | : i = 1, . . . ,m}.

2. Rm 1 : Rm R

x1 := |x1|+ + |xm|=m

i=1

|xi |.

R. (Rm, 1) `m1 .

3. Rm 2 : Rm R

x2 :=

(m

i=1

|xi |2)1/2

.

,

( CauchySchwarz)

x1, . . . ,xm y1, . . . ,ym .

m

i=1

|xiyi |

(m

i=1

|xi |2)1/2(

m

i=1

|yi |2)1/2

.

:

(x ,y) =m

i=1

xiyi

x22 = (x ,x).

x + y22 = (x + y ,x + y) = (x ,x) + (x ,y) + (y ,x) + (y ,y) x22 + 2|(x ,y)|+y22cs x22 + 2xy+y22 = (x+y)2

. x + y2 x+y .

4. , Rm p-,1 < p < ,

xp :=

(m

i=1

|xi |p)1/p

.

( Holder)

x1, . . . ,xm y1, . . . ,ym p,q > 1 1 1p +

1q = 1,

m

i=1

|xiyi |

(m

i=1

|xi |p)1/p(

m

i=1

|yi |q)1/q

.

CauchySchwarz Holder p = q = 2.1 p q .

1. ` `(N) x : N R . ` supremum : ` R

x := sup{|x(n)| : n = 1,2, . . .}.

2. c0 c0(N) ,

c0 ={

x : N R limn

x(n) = 0}

( `) . supremum `.

3. `1 `1(N) 1- ( ):

`1 =

{x : N R

n=1

|x(n)|< +

}.

c0( n=1 |x(n)|< + limn x(n) = 0). 1 : `1 R

x1 :=

n=1

|x(n)|.

4. 1 p < , `p `p(N) p-:

`p =

{x : N R

n=1

|x(n)|p < +

}.

p

xp :=

(

n=1

|x(n)|p)1/p

.

Minkowski p .5. c00 c00(N) :x c00 n0 n0(x) N x(n) = 0 n n0. p, 1 p .

1. C ([0,1]) [0,1]:

C ([0,1]) = {f : [0,1] R | f } .

. C ([0,1]) : C ([0,1]) R,

f = sup{|f (t)| : t [0,1]}.

sup , |f | : [0,1] R , max , , .

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

f := f +f .

f d := f

( ) C 1([0,1]).

4. f ,g : [0,1] R ,

dp(f ,g) =

( 10|f (t)g(t)|pdt

)1/p 1 p <

d(f ,g) = max{|f (t)g(t)| : t [0,1]}.

(X ,) . X x : N X . xn := x(n) n- x {xn}n=1 {xn} (xn) x = (x1,x2, . . . ,xn, . . .).

: (an) a R:

an a > 0 no N : n no |ana|< .

(xn) (X ,) x X ( - x)

> 0 n0 n0() N n n0 (xn,x) < .

xn x xn x .

(xn) x :

xn x ( > 0 no N : n no (xn,x) < .)

> 0 no N n no xn B (x ,) {y X : (y ,x) < }.

> 0 (xno ,xno+1,xno+2, . . .) B (x ,).

> 0

P() : {n N : (xn,x) } .

(xn) x :

> 0, P():

P() : {n N : (xn,x) } .

(xn,xn+1,xn+2, . . .) B (x ,), xm (xn,xn+1,xn+2, . . .) B (x ,)

: (xn) x > 0 n N m n (xm,x) .

xn 9 x ( > 0 n N m n : (xm,x) .)

(X ,) .

(xn) (X ,) x X . xn x

((xn,x))n 0 R.

(xn) (X ,). (xn), .

(xn),(yn) X x ,y X xn x

yn y , (xn,yn) (x ,y).

:

|(xn,yn)(x ,y)| (xn,x) + (yn,y).

: (xn) n0 N xn = xn0 n n0. , ( xn0).

1. X (xn) X . (xn) (X , ) . Hamming (HN ,h) .(3)

2. . (X1,d1), . . . ,(Xk ,dk) X = ki=1 Xi . X d = ki=1 di ,

d(x ,y) =k

i=1

di (x(i),y(i)),

x = (x(1), . . . ,x(k)), y = (y(1), . . . ,y(k)) x(i),y(i) Xi ., X :2

xn = (xn(i))n

x = (x(i)) i = 1,2, . . . ,k, xn(i)n

x(i)

2 d .

3. Hilbert H

[1,1]N =

{x : N R

|x(n)| 1,n = 1,2, . . .}

d(x ,y) =

n=1

|x(n)y(n)|2n

,

x = (x(n)) y = (y(n)). ([1,1]N,d) Hilbert H . .

(xm) H :

x1 = (x1(1),x1(2), . . . ,x1(n), . . .)

x2 = (x2(1),x2(2), . . . ,x2(n), . . .)...

xm = (xm(1),xm(2), . . . ,xm(n), . . .)...

4. .

(Xn,dn), n = 1,2, . . . dn(x ,y) 1 x ,y Xn n = 1,2, . . .. X = i=1 Xi d : X X R :

d(x ,y) =

n=1

1

2ndn(x(n),y(n)), (1)

x = (x(1),x(2), . . .), y = (y(1),y(2), . . .) x(n),y(n) Xn n = 1,2, . . .. d : X .

5. (C [0,1], )

d(f ,g) = sup{|f (t)g(t)| : t [0,1]} . : (fn)

fn(t) =

2n3t 0 t 1

2n2

2n3( 1n2 t) 1

2n2< t 1

n2

0 1n2

< t 1

( ). t [0,1] lim

nfn(t) = 0,

d(fn,0)+, d(fn,0) = max{|fn(t) : t [0,1]}= n n.

6. (C [0,1], 1)

d1(f ,g) = 10|f (t)g(t)|dt

. (fn)

fn(t) = tn, t [0,1].

d1(fn,0) =1

n + 1 0 t [0,1]

limn

fn(t) = 0, fn(1) = 1 n.

(fn)

f (t) =

{0 0 t < 11 t = 1

C [0,1].

( )

(xn) (X ,). (xn) ( Cauchy) > 0 n0 n0() N m,n n0 (xm,xn) < .

(X ,) . , X Cauchy.

( )

(xn) (X ,). (xn) A = {xn : n N} X . , C > 0 (xm,xn) C m,n N.

(X ,) . , X ., X .

: !! .. xn = (1)n (R, | |).

;

(Q, | |) : qn = (1 + 1n )n

(R,) (x ,y) = x|x |+1 y|y |+1 : xn = n.

(R, | |) .

; .

(xn) (X ,). k1 < k2 < < kn < (xkn) (xn).

: xn =1n yn =

12n

zn =1

n+5 wn =1pn

pn = n- .

() k : N N x : N X X , x k : N X (xn). (xn) (xn) .

() (kn) , kn n n = 1,2, . . .. :.

xn x (xkn) (xn) ,

: xkn x

(xn) x > 0 (xkn) (xn) n N (xkn ,x) .

: xn 9 x > 0 {n N : (xn,x) } k1 < k2 < k3 < .. .

(xkn ,x) n N.

(X ,) (xn) X . (xn) Cauchy , .

BolzanoWeierstrass

(BolzanoWeierstrass)

Rm ( ) .

BolzanoWeierstrass :

(c0, ) supremum : x = (x(n)) y = (y(n))

d(x ,y) = sup{|x(n)y(n)| : n = 1,2, . . .}.

(en) e1 = (1,0,0,0, . . .)

e2 = (0,1,0,0, . . .)

e3 = (0,0,1,0, . . .)...

d(en,em) = en em = 1 n 6= m. (en) : 1.

: (C [0,2], 1)

: d1(f ,g) = 20|f (t)g(t)|dt.

(fn)

fn(t) =

tn 0 t 1

1 1 < t 2

d1(fn, fm) =1

n+1 1

m+1 m n, f C ([0,2]) lim

nd1(fn, f ) = 0

f f (t) = 0 0 t < 1 f (t) = 1 1 t 2, f . (fn) , , .

: A R , f : A R x0 A, f x0 : > 0 > 0 :

x A |xx0|< , |f (x) f (x0)|< .

(X ,) (Y ,) . f : X Y x0 X

> 0 (x0,) > 0 : x X (x ,x0) < (f (x), f (x0)) < .

f : X Y X X . f : (X ,) (Y ,) C (X ,Y ). Y = R C (X ) C (X ,R).

() X (Y ,) . f : (X , ) (Y ,) .

() f : N X .() I : (c00, ) (c00, 2) .

f : X Y x0 X > 0 : > 0 x X (x ,x0) < (f (x), f (x0)) .

, R.

( )

(X ,) (Y ,) f : X Y x0 X . :() f x0.

() (xn) X xn x0

f (xn) f (x0).

() (xn) xn x0, (f (xn))

-.

:

(X ,) x0 X .() - x0 > 0

B (x0,) = {x X : (x ,x0) < }.

() - x0 > 0

B (x0,) = {x X : (x ,x0) }.

() - x0 > 0

S (x0,) = {x X : (x ,x0) = }.

() X , r > 0

B(x , r) =

{{x}, r 1X , r > 1

S(x , r) =

{X \{x}, r = 1/0, r 6= 1. .

() R ,

B(x ,) = (x ,x + ), B(x ,) = [x ,x + ],S(x ,) = {x ,x + }.

() R2, B1(0,) B2(0,) B(0,).B2(0,) 0 B1(0,) B(0,) B2(0,) (1,0),(1,1),(1,0),(11).

:

(X ,) A X . x A (interior point) A x > 0 B (x ,x) A.

(X ,) G X . G - (open) x G x > 0 B (x ,x) G . , G .

A X x A > 0 B (x ,)Ac 6= /0.

() .

() .

() (a,b] (R, | |), a < b .

() (R, | |), Q , . Qc ...

: X , /0, ,

(X ,) . , :() X /0 .

() (Gi )iI X

iI Gi .

() G1,G2, . . . ,Gn ni=1 Gi = G1 Gn .

( (X ,) X .)

(X ,) ., R Gn = (1, 1n ), n N,

n=1 Gn = (1,0]: .

(X ,) . :() G X .

() x G (xn) X xn

x n0 N n n0 xn G .

(X ,) . V X ( ) X .

( )

U (R, | |) () .

(X ,) F X . F - (closed) F c X \F -.

() (X ,) {x}, x X .

() (X ,), B(x , r) ., R , [a,b] .

() Q R , R\Q ( ).

() ( !).

() (X ,d) . x X (xn) X , xn x .

E = {xn : n = 1,2, . . .}{x}

(X ,d).

. , .

: X , /0, ,

(X ,) . :() X , /0 .

() F1,F2, . . . ,Fn X ,

ni=1 Fi

.

() (Ei )iI X ,

iI Ei .

. , (R, | |), Fn =

[1n ,1], n = 2,3, . . .,

n=1 Fn = (0,1]

.

(X ,) , F X .() F .

() F

F . , (xn) F

xn x X x F .

(): xn F n (xn), lim

nxn F .

:

(X ,) , F X .() F .

() x X : > 0

B(x ,)F 6= /0 F .

() F F . , (xn) F

xn x X x F .

() (): F F c , .

x F c = > 0 .. B(x ,) F c ,. B(x ,)F = /0

B(x ,)F 6= /0 > 0 = x F .

() = (): (xn) .. xn x xn F n.... x F : > 0 n0 xn0 B(x ,), B(x ,)F 6= /0. (), x F .() = (): x X > 0 B(x ,)F 6= /0. ... x F . n, B(x , 1n )F 6= /0: xn B(x ,

1n )F .

n, xn F (xn,x) < 1n , xn x . (), x F .

A (X ,). ( interior) A A intA ( A). ,

A intA = {x A | x > 0 : B(x ,x) A} .

() x A Vx x Vx A() A X A A .

() (a,b] R (a,b).

() Q R /0.() .

A,B (X ,). :() A A.() A =

{V A : V }. , A

A.

() A = A A .

() A B , A B.() (AB) = AB.() AB (AB).

: (R, | |), A = [0,1], B = (1,2) AB = (0,1) (1,2) ,(AB) = (0,2). A = Q B = R\Q AB = /0, (AB) = R.

(X ,) A X . x X A > 0 AB(x ,) 6= /0( x A).

( x A, .)

() (R, | |) A = (0,1]. 0 A 0 / A. 1 A 1 A.() (xn) (X ,) xn

x , x A = {xn : n N}.

() X , A X , x X A x A.

(X ,) A X . x X A (an)

A an x .

(X ,) A X , (closure)A ( cl(A)) A .,

A cl(A) = {x X : > 0,AB(x ,) 6= /0}.

A X A A .

() (R, | |) Q = R R\Q = R.() (R, | |), a,b R a < b cl(a,b) = cl(a,b] = cl [a,b) = [a,b].() X , A X cl (A) = A = A.

(X ,) A,B X . :() A A.() A =

{F A : F }. , A

X A.

() A = A A .

() A B , A B .) AB = AB .() AB AB .

.: (R, | |) QQc = /0 QQc = R.

(X ,), A X :

A = {x X | > 0 : B (x ,) A}

: A.

A = {x X |(xn) xn A xn x}

= {x X | > 0 : B (x ,)A 6= /0}

: A.

G X G = G

F X F = F

G ,F ,G A F = G A A A F

(X ,) , A,B X . :() X \A = (X \A)

() X \B = (X \B)

. () x X . : > 0 B(x ,)A 6= /0, x A. > 0 B(x ,)A = /0, B(x ,) X \A, x (X \A). A (X \A) X . X \A = (X \A). (), () B = X \A.

( )

(X ,) A X . x X (accumulation point) A x A x . , > 0

B(x ,) (A\{x}) 6= /0.

A A A.

A X x X , :() x A.() > 0 AB(x ,) .() (an) A an

x an 6= x n N.

() (X ,) A X . , A = AA., A .() A ().

(X ,) A X . x X (boundary point) A x A Ac ., > 0 B(x ,)A 6= /0 B(x ,)Ac 6= /0. A (boundary) A bd(A) (A).

()

(X ,) A X . ,() bd(A) = bd(Ac). () A = bd(A) int(A) ( ).() X = int(A)bd(A) int(Ac) ( ).() bd(A) = A\A. , bd(A) = AX \A. , bd(A) .() A bd(A) A.

A 6= /0 (X ,), A(x ,y) = (x ,y), x ,y A A. x A BA(x , r) = B (x , r)A.

(X ,) A X . :() G A .. (A,A) (. A-) - V X G = AV .() F A A- - E X F = AE .

() (R, | |) A = (0,1]{2}. (0,1] {2} A.() (R3, 2) xy - H ( (x ,y ,0)). D xy -(D = {(x ,y ,0) R3 : x2 + y2 < 1}) H R3.

: , ,

()

A,B . A,B f : A B , 1-1 (f : A B). A =c B |A|= |B| A B .

1. f : (0,1) (0,2) f (x) = 2x .f : (0,1) (a,b) f (t) = (1 t)a + tb.2. N A (:) n 7 2n.

3. f : Z N f (n) ={

2n, n 02|n|1, n < 0

4. f : [0,1] [0,1) f (x) ={

1n+1 , x M x =

1n

x , x /M(M = { 1n ,n N})

A,B,C . () A A,() A B , B A () A B B C , A C .

A A = /0 n N f : A {1,2, . . . ,n} 1-1 . , A n A n .3

|A|= n. A .

3 0.

A . () A .() 1-1 f : N A, B A B N.

1. Z , N Z.2. NQ.3. R .

.

A 1-1 f : N A, A N., A .

. 0 ( 0). , A |A|= 0.

() Z .() NN = {(m,n) : m,n N} .() A,B , AB = {(a,b) : a A,b B} .

A . :() A .() f : NA, .() g : A N, 1-1.

B N , B N.

Q . U (R, | |) .

(Cantor, 1899)

(Ai )iI X . I ,

iI Ai

.

;

[0,1] = {x R : 0 x 1} .

: R ( ). , Q [0,1] .

R R\Q .

(Cantor)

2N = {0,1}N = {x = (x(n)) : x(n) {0,1}, n = 1,2, . . .}

.

P(A) A .

: , ,

A,B (A B) f : A B , 1-1 (f : A B).

A A = /0 n N : A {1,2, . . . ,n}.

A A N. [. A = Q.]

A A N.

A [..A = [0,1] {0,1}N].

(X ,) D X . D (dense) X , D = X .

() Q R\Q (R, | |).() c00 (`1, 1).() ( Kronecker). 2 R\Q.

D() := {(cos(n),sin(n)) : n N}

S1 = {(x ,y) R2 : x2 + y2 = 1}.

(X ,) D X . :

() D X .

() F D F , F = X .() G X G D 6= /0.() x X > 0 D B(x ,) 6= /0.() x X (xn) D xn

x .() (X \D) = /0.

(X ,) ( separable) . , D = {d1,d2, . . .} X D = X .

() Rd , p, . Qd .() `p, 1 p < .() ` [: ].

() Hilbert, H .

(X ,) . :() X .() O X : G X x G U O x U G .:

(X ,) : > 0 {X n : n N} X diam(X n ) X =

n=1

X n .

(X ,) . (A,A) X .

(X ,) .

(X ,) {B(xi , ri ) : i I} , .

X A : > 0 (x ,y) x ,y A, x 6= y . .

() ` (`(N), ) .() (.. = R), (, ) (, ) .

f : (X ,) (Y ,) x0 X : > 0 (x0,) > 0 : x X (x ,x0) < (f (x), f (x0)) < .

> 0 (x0,) > 0 :

f (B (x0, )) B (f (x0),).

: f : X Y :

A X : f (A) = {f (x)|x A}= {y Y |x A : f (x) = y}B Y : f 1(B) = {x X |f (x) B}.

( f : X Y X x X .)

f : (X ,) (Y ,). :() f .

() G Y , f 1(G ) X .

() F Y , f 1(F ) X .

f : (X ,) (Y ,). :() f .

() A X f (A) f (A).() B Y f 1(B) f 1(B).() C Y f 1(C ) (f 1(C )).

Urysohn

(Urysohn)

(X ,) A,B X AB = /0. , f : X [0,1] :() f (x) = 0 x A.() f (x) = 1 x B .

f (x) =dist(x ,A)

dist(x ,A) + dist(x ,B)

Urysohn

A,B (X ,).() A,B G ,H A G , B H G H = /0.() A,B G ,H A G , B H G H = /0.

(X ,) E ,F X . E ,F .

f : (X ,) (Y ,) : > 0 = () > 0 : x ,y X (x ,y) < (f (x), f (y)) < .

f : x X > 0 = (x ,) > 0 y X (x ,y) < (f (x), f (y)) < .

. . : p : R R p(x) = x2 : > 0, x =

1 y =

1 +

2 , |x y |<

|p(x )p(y )|= 1 + 2

4> 1.

:

f : (X ,) (Y ,) : > 0 = () > 0 : x ,y X (x ,y) < (f (x), f (y)) < .

, > 0 > 0 x X :f (B (x , )) B (f (x),)., > 0 > 0 A X :diam (A) < diam (f (A)) < .

() f : (X , ) (Y ,) . a : N Y .() (X ,) , A X dA : X R : t 7 dist(t,A) inf{(t,a) : a A}. dA .: t,s X , |dA(t)dA(s)| (t,s).() f : R R , ( , . 6).

() f : RR , limx |f (x)|= 0, .

(X ,),(Y ,) f : X Y . () f . () (xn),(zn) X (xn,zn) 0, (f (xn), f (zn)) 0.

f : (X ,) (Y ,). :() f .() f - -.() f .

: () ; (): .. f (x) = 1x ((0,+), | |) () ; (): .. g(x) = x2 (R, | |).

Lipschitz

f : (X ,) (Y ,) C > 0. f CLipschitz x ,y X

(f (x), f (y)) C (x ,y).

f Lipschitz Lipschitz C > 0.

() Lipschitz . : .. f (x) =

x (R+, | |).

() f : R R Lipschitz.

() X . : (X , ) (R, | |) 1Lipschitz, .

Lipschitz

f : (X ,) (Y ,) Lipschitz. f X Y .

. ..

id : (R, ) (R, | |).

()

(X ,),(Y ,) . f : X Y (isometry) ,

(f (x), f (y)) = (x ,y)

x ,y X .

() 1-1 .

() Lipschitz.

() : (shiftoperator) Sr : `2 `2 Sr (x1,x2,x3, . . .) = (0,x1,x2, . . .) .

X , X . ( ) .

xn x xn

x .

X , X . :

() , .() id : (X ,) (X ,) id1 .

() ( Hausdorff) > 0 x X 1,2 > 0 B (x ,1) B (x ,) B (x ,2) B (x ,).() G X .() F X .

X , X .

f : (X ,) (Y ,). f (homeomorphism) 1-1, . (X ,) (Y ,) f : (X ,) (Y ,). X

hom Y X ' Y .

() .

() X . , , (X ,) (X ,) . .

f : (X ,) (Y ,) 1-1 . :

() f .

() (xn) X x X , xn x

f (xn) f (x).

() G X f (G ) Y .() F X f (F ) Y .() d(x ,y) = (f (x), f (y)) X .

(X ,) .

:

f : (X ,) (Y ,).

f : 1-1, , f 1 : (Y ,) (X ,) . X ' Y () f : (X ,) (Y ,). ( X = Y ) () id : (X ,) (X ,) : x x.

( C > 0), f CLipschitz x ,y X

(f (x), f (y)) C (x ,y).

f x ,y X

(f (x), f (y)) = (x ,y).

() R 6' Z, R 6'Q

() Z 6'Q() (2 ,

2 ) R ( )

tan : (2 ,2 ) R,

diam((2 ,2 )) = diam(R) = .

() (0,1)' (a,b), [0,1)' [a,b)' (c,d ] [0,1]' [a,b].

() (0,1) (a,b) (c ,d)( b c).

() (0,1) [0,1).

(X ,) (complete) (xn) X .

, (xn) :

> 0 n0 N : m,n n0 (xn,xm) <

x X (xn,x) 0.

( )

() X , (X , ) .() (R, | |) ( ).() (Rm,2), 2 , .

( )

() {(Xi ,di )}ki=1 . (ki=1 Xi ,

ki=1 di ) (Xi ,di )

i = 1,2, . . . ,k .

() {(Xn,dn)}n=1 dn(x ,y) 1 x ,y Xn,n = 1,2, . . . . X = n=1 Xn

d(x ,y) =

n=1

1

2ndn(x(n),y(n))

x = (x(1), . . . ,x(n), . . .) X . Xn (X ,d) .

( )

() (Q, | |) .() (R,) (x ,y) = |arctanxarctany | : xn = n -, -. .

Banach

(X , ) . X Banach , (X ,d) d(x ,y) = xy .

. `() x : R,

d(x ,y) = sup{|x(i)y(i)| : i }

.

` d .

(X ,) F X . (F ,|F ) , F X .

(X ,) F X . F X (F ,|F ) .

:

(X ,) F X . F X (F ,|F ) .

c0 d ( `) .

C ([a,b]) = {f : [a,b] R|} d( `([a,b])) .

`().

Cantor

: (R, | |).

(CantorFrechet)

(X ,) . : {Fn}nN , X diam(Fn) 0,

n=1 Fn 6= /0.

, x X

n=1 Fn = {x}.

: : (R, | |): Fn = (0, 1n ) : , diam(Fn) 0,

n Fn = /0 :

!

Fn = [n,) : , ,

n Fn = /0 :diam(Fn) 6 0.

Cantor

()

(X ,) {Fn}nN , X diam(Fn) 0

n=1 Fn 6= /0,

(X ,) .

, .. (Qc , | |): Fn = [ 1n ,

1n ] : , diam(Fn) 0,

n Fn = /0.

Cantor ()

(X ,) {An} X diam(An) 0.,

n=1 An .

(X ,) (xn) X . (xn):

Rn = {xk : k n}, n = 1,2, . . .

(X ,) (xn):() (xn) .

() diam(Rn) 0 n .

Baire

G1,G2, . . . ,Gm (X ,), mi=1 Gi ( , ). [ ]

: .: (Q, | |) (qn) Q Gn = Q\{qn}. Gn Q, .

(Baire)

(X ,) . (Gn) X ,

n=1 Gn 6= /0.

, X .

:

(X ,) . (Fn) X

n=1 Fn = X ,

k N int(Fk) 6= /0.

Baire:

V X . ... V (

n=1 Gn) 6= /0.G 1 = X V G1 6= /0 ,

B(x1, r1) V G1 0 < r1 < 1.

G 2 = X G2B(x1, r1) 6= /0

B(x2, r2)G2B(x1, r1)V G1G2 0< r2 < 1/2.

: n B(xn, rn):

diam(B(xn, rn)) 2nB(xn, rn) GnB(xn1, rn1), B(xn, rn) B(xn1, rn1)

B(xn, rn) V G1 Gn.

Cantor: x X x

n=1 B(xn, rn). x V G1 . . .Gn n N.

G F

(X ,) A X .() A G X .() A F X .

(R, | |): () (a,b] F G : k N, :

(a,b] =

n=k

[a +

1

n,b

]=

n=1

(a,b +

1

n

).

() Q F (.) G (.: !)

Baire

G G R. , G .

Q G R.

... .

Baire

(X ,) .() A X int(A) = /0. ( int(A) = /0: .. Q (R, | |).)() B X (X ) X , En, n = 1,2, . . . X , B =

n=1 En.

() C X ( X ) .

Baire :

( ).

f : (X ,) (Y ,) . x X , f x :

f (x) =inf{diam(f (B (x , ))) : > 0}=lim

0diam(f (B (x , ))) [0,+].

f : (X ,) (Y ,) x X . :() f x .() f (x) = 0.

C (f ) f : (X ,) (Y ,) G X .

D(f ) F :

D(f ) =

n=1

{x X : f (x)1

n}

: f : R R (. C (f ) = R\Q).( [0,1] : f (mn ) =

1n (m,n) = 1 f (x) = 0

x /Q).

f : R R C (f ) = Q.

... Q G .

Banach

( )

f : X X x0 X . x0 f f (x0) = x0. Fix(f ) f .

f , Fix(f ) X .

(Banach)

(X ,) T : X X : 0 < c < 1

(T (x),T (y)) c (x ,y)

x ,y X . , z X T (z) = z .

Banach

() n- :

(T n(x),z) cn

1 c(x ,T (x)).

() T : R R T (x) = log(1 + ex) |T (x)T (y)|< |xy | x ,y R, .

() (X ,) .: f : (0,1) (0,1) f (x) = x2 |f (x) f (y)| 67 |xy | x ,y (0,1), f .

( )

(X ,) . (Y ,) X T : X Y T (X ) Y .

( )

(X ,) . , (X , ) T : X X T (X ) X .

(X ,) . T : (X ,) (`(X ),d).

T : x fx fx(t) = (t,x)(t,a), t X .

( )

(X1,1) (X2,2) (X ,). , : X1 X2, , (T1(x)) = T2(x) x X .

X1 X2

T1(X ) T2(X ) T1 T2

Xi X

( .)

( )

(X ,) A X . U = {Ui}iI X A, A

iI Ui .

J I A

iJ Ui , {Ui}iJ U U = {Ui}iI A.

()

(X ,) (compact) X . , :

U = {Ui}iI X X =

iI Ui

Ui1 , . . . ,Uim U X = Ui1 Uim .

()

K (X ,) , k .

K (X ,) (Vi )iI K - X , (Vij )

mj=1 (Vi )iI

: K m

j=1 Vij .

() X , (X , ) X .() (R, | |) .

() S` = {x = (x(n)) ` : x = 1} `.

() (X ,) , x X (xn) X xn x , K = {xn : n = 1,2, . . .}{x} X .

(X ,) K X . , K X .

: ().

(X ,) F X . , F .

( )

(X ,) (sequentially compact) (xn) X (xkn) x X .

( )

(X ,) (totallybounded) , . > 0 m N x1, . . . ,xm X

X =mi=1

B(xi ,).

, A X > 0 x1, . . . ,xm X

Ami=1

B(xi ,).

1. A X , B A .2. xi A (A 6= /0).

() (R, | |) ( (0,1) ).

() X , (X , ) X .

() 4 (S` ,d) , .

4S` = {x = (x(n)) ` : x = 1}

( )

(X ,) . :1 (X ,) .2 A X

X (, A 6= /0).3 X .

4 X .

(X1,d1),(X2,d2) . A X1, B X2 , AB X1X2 X1X2.

:

(Xi ,di )mi=1 . X = mi=1 Xi

Xi d X , (X ,d) .

K X , . : (`2, 2), B(0,1) , : (en)n .

Rm, m 1, . K Rm .

., .

( )

(Fi )iI X 6= /0 J I

iJFi 6= /0.

: () {(,x ] : x R} R. () A N N\A .

(X ,) Fi X (Fi )iI ,

iIFi 6= /0.

f : (X ,) (Y ,) .(X ,) = f .

f : (X ,) (Y ,) .K X = f (K ) Y .

f : (X ,) (Y ,) , 11 .(X ,) = f .. : (X ,) . , f : [0,1) [2,3] [0,2] f (x) = x 0 x < 1 f (x) = x1 2 x 3. f , 1-1, , f 1 ( y = 1).

f : (X ,) (Y ,) (X ,) , f (X ) Y .

(X ,) f : X R . , f .

:

x R+. (xn).; , ; n N,

gn : [0,1] R gn(x) = xn.

(gn). x [0,1], (gn(x)) . x ;

x R, x 6= 1 :(1x)(1 + x + x2 + + xn) = 1xn+1

1 + x + x2 + + xn = 11x

xn+1

1x

, |x |< 1, 1 + x + x2 + + xn n 11x

|x | 1 1 + x + x2 + + xn + . . . . fn(x) = 1 + x + x2 + + xn, (n N), f (x) =

1

1x.

; ;

( )

X , (Y ,) , fn, f : X Y , n N.

( )

(fn) (pointwise) f : x X (fn(x)) f (x) Y , .limn

(fn(x), f (x)) = 0. : x X > 0 n0 = n0(x ,) N n n0 (fn(x), f (x)) < .

X , fn, f ,gn,g : X R. fnk.s. f

gnk.s. g , : (i) a,b R afn + bgn

k.s. af + bg (ii) fngn

k.s. fg .

(1 )

(X ,) fn, f : X R. fn f fn , f ;

: .. fn : [0,1] R : fn(t) = tn : .

limn

fn(t) =

{0, 0 t < 11, t = 1

.

(2 )

fn, f : [a,b] R. fnk.s. f fn

Riemann [a,b], () f R [a,b];

() ( R-. ) ba

fn(x)dx ba

f (x)dx ;

() : .. Q = {qn : n N}, fn = {q1,...,qn}k.s. f = Q,

f Riemann [0,1].() : .. fn, f : [0,1] R : fn(t) = n2t(1 t)n, f (t) = 0.

(3 )

I R fn, f : I R. fn f fn I , () f I ;() ( ) f n f ;

() : fn(t) = tn [0,1], f f (t) = limn fn(t) .() : .. () fn, f : [0,1] R : fn(t) = t1+nt , f (t) = 0... () gn,g : (0,) R : gn(t) = sin(nt)n , g(t) = 0.

( )

() fn : [0,1] R : fn(t) = tn

limn

(limt1

fn(t)) = 1 6= 0 = limt1

( limn

fn(t)).

limn fn(t) = f (t) =

{0, 0 t < 11, t = 1.

() fn : [0,1] R fn(t) ={

0, 0 t 1/nsin(/t), 1/n t 1

limn(limt0 fn(t)) = 0 limt0(limn fn(t)) .

limn

fn(t) = f (t) =

{0, t = 0sin(/t), 0 < t 1

( )

X , (Y ,) , fn, f : X Y , n N. ( )

(fn) (pointwise) f : x X (fn(x)) f (x) Y , .limn

(fn(x), f (x)) = 0. : x X > 0 n0 = n0(x ,) N n n0 (fn(x), f (x)) < .

( )

(fn) (uniformly) f : > 0 n0 = n0() N : n n0 x X (fn(x), f (x)) < .

fnom. f fn

k.s. f

fnk.s. f ; fn

om. f .. fn(t) = tn, t (0,1)

sup{tn : t (0,1)}= 1 n.

:

X , (Y ,) , fn, f : X Y , n N.

( )

(fn) (uniformly) f : > 0 n0 = n0() N : n n0 x X (fn(x), f (x)) < .

(Y ,) = (R, | |): (fn) f > 0 n0 = n0() N : n n0 ( f fn ) fn f < .

: n n0, fn 2 f .

: (fn) fn : X R.1 f fn f ;

x X (fn(x)): . x X lim

nfn(x) , f : X R

f (x) = limn

fn(x).

2 fn f : fn f fn f [0,+]: fn

om. f (fn f ) 0 n .

() (X ,d) , (xn) X xn x ,fn, f : X R fn(t) = d(t,xn) f (t) = d(t,x) t X . : |fn(t) f (t)| d(xn,x) . t.() fn : R R fn(x) = xn . fn 0 .,

fn0 = sup{|x |n

: x R}

= +

n N. .() fn, f : [0,M] R fn(t) =

(1 + tn

)n, f (t) = et .

: t 0, fn(t) f (t).

f fn?= eM

(1 +

M

n

)n lim

nf fn = 0.

() hn,h : [0,+) R hn(t) =(1 + tn

)n, h(t) = et .

hhn en(

1 +n

n

)n= en2n+.

() fn : [0,) R fn(x) = nx+n2 . x 0

|fn(x)|=n

x + n2 n

n2=

1

n: .

() fn : [0,1] R fn(x) = xn. fn f ,

f (x) =

{0, 0 x < 11, x = 1.

.

() fn : [0,1] R fn(x) ={

11+nx ,

1n x 1

nx2 , 0 x 0 n0 = n0() : n,m n0 fn fm < .

:

fn f 0 > 0 n0 N : n n0 fn f < /2 n,m n0 fn fm <

: gn = fn fn0 (n n0) gn `(X ) (gn) , ( (`(X ), ) ) g `(X ) gng 0, f = g + fn0 fn

om f .

(X ,d) , fn : X R f : X R fn

o f . , x0 X (xn) X xn x0 fn(xn) f (x0).

: |fn(xn) f (x0)| fn f + |f (xn) f (x0)|.

(Dini)

(X ,d) fn : X R , f : X R. , fn

om f .

: > 0, Kn() = {x X : |fn(x) f (x)| }. Kn() Kn+1()

n Kn() = /0.

(X ,) , f , fn : X R x0 X . :1 fn f X , 2 fn x0.

, f x0., fn X , f X .

, fn , fn f f , .

(fn) [a,b] R. fn : [a,b] R Riemann [a,b] fn f [a,b]. ,() f Riemann [a,b]

() ba

fn(x)dx ba

f (x)dx .

( () .)

: (fn) (0,) fn(x) = sin(nx)nom 0 (f n)

.v.

fn, f : [a,b] R fn f . () fn [a,b]() f n [a,b]() (f n) [a,b]. f [a,b] f = lim f n.

: fn f [a,b].

:

fn,g : [a,b] R, n N. () fn [a,b]() f n [a,b]() f n g [a,b], () x0 [a,b] (fn(x0)) ( R). (fn) f : [a,b] R, f [a,b] f = g .

fk : X R, k N. n N sn : X R

sn(x) = f1(x) + f2(x) + + fn(x).

s : X R snk.s. s X ,

k=1 fk s X

s =

k=1

fk .

snom s, k=1 fk

s X .

()

k=0

xk .

k=0

xk =1

1x x (1,1).

sup{|sn(x) s(x)| : x (1,1)}= +

n N: .

()

k=0

xk

k!= ex x R.

K R, R.

k=1 fk s X , s X .

fk ,gk : X R, k N a,b R. k=1 fk = f k=1 gk = g X ,

k=1(afk + bgk) = af + bg

X . .

( Cauchy)

fk : X R, k N. k=1 fk ( ) X : > 0 n0 = n0() N : n > m n0 x X ,

|fm+1(x) + + fn(x)| .

( Weierstrass)

fk : X R , k N.

sup{|fk(x)| : x X} Mk , k N

Mk |fk |,

k=1

Mk < +.

, k=1 fk X .

k=1 fk k=1 fk.

k=1sin(kx)

k2, x R.

R, .

(X ,) , f , fk : X R x0 X . 1 k=1 fk f X , 2 fk x0.

f x0., fk X , f X .

(fk) [a,b]. fk : [a,b] R Riemann k=1 fk f [a,b]. f Riemann- [a,b] b

a

(

k=1

fk(x)

)dx =

k=1

( ba

fk(x)

)dx .

(Weierstrass, 1872): f (x) =k=1

12k

cos(10kx)

!

fk , f : [a,b] R k=1

fk = f .

() fk [a,b]() f k [a,b]

() k=1

f k [a,b]

f [a,b] (

k=1

fk

)=

k=1

f k .

k=1

fk f .

:

fk : [a,b] R, k N. () fk [a,b]() f k [a,b]() k=1 f k [a,b], () x [a,b] k=1fk(x) ( R). k=1 fk f : [a,b] R, f [a,b] (

k=1

fk

)=

k=1

f k .

C (K ) (K ,d)

`(K ) f : K R f = sup{|f (x)| : x K} .

fn f 0 fn f K .

(C (K ), ) (`(K ), ).( = .)

(C (K ), ) (`(K ), ).( .)

(K ,d) . (C (K ), ) .

Weierstrass

(Weierstrass)

f : [a,b] R . > 0 p : R R p [a,b]

f p .

, x [a,b],

|f (x)p(x)| .

:

(Weierstrass)

f : [a,b] R . (pn) pn f [a,b].

Metric SpacesMetrics in linear spacesFinite dimensional spacesSequence spacesFunction spaces

Convergence of sequencesConvergent sequencesExamples of convergenceCauchy sequences and bounded sequencesSubsequences

ContinuityTopology of metric spacesOpen sets

Closed setsInterior of a setContact pointsClosure of a setClosure and InteriorAccumulation points and the boundaryRelatively open, relatively closed sets

Finite, infinite, uncountableCountable and uncountable setsDense subsets

Separable metric spacesContinuous functionsUrysohn's Lemma

Uniformly continuous functionsLipschitz functions

Isometries, homeomorphisms, equivalent metricsIsometriesEquivalent metricsHomeomorphisms

Complete metric spacesCantor's TheoremThe Baire Category TheoremGd and Fs setsOscillation and points of continuity

Banach's fixed point theoremCompletion of a metric spaceCompactnessCharacterisations of compactnessBasic properties of compact setsContinuous functions on compact setsSequences of functionsSequnces of functions: Uniform convergenceUniform convergence criteriaContinuity, integral, derivative

Series of functionsThe space C(K) where (K,d) is a compact metric spaceThe Weierstrass Approximation Theorem

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