матан Бесов - весь 2012

cos kx sin kx

!" #

%$ $ {eikx}∞k=−∞ "

' & '

" $

( & R $

& " x y

" )

ρ(x, y) 0 $ * + & x y ' '

#

,◦ ρ(x, y) = 0 x = y-

.◦ ρ(x, y) = ρ(y, x) * +-

/◦ ρ(x, y) ρ(x, z) + ρ(z, y) * +

'' $' &

!$ $' &

Rn x = (x1, . . . , xn) xi R *i = 1 n+

§ ,0 ,

& C([a, b])

$ $ [a, b] ) "

ρ(f, g) = max |f (t) − g(t)|.

a t b

ε

! R

" # "

#

x, y R x + y R

x R $% λ

&

'◦ x + y = y + x x, y R(

)◦ (x + y) + z = x + (y + z) x, y, z R(

R(

+◦ x R R

% −x x +

−

+ ( x) = 0(

,◦ (λμ)x = λ(μx) x R λ, μ R( -◦ 1x = x x R(

.◦ (λ + μ)x = λx + μx x R λ, μ R( /◦ λ(x + y) = λx + λy x, y R λ R

0 % $

x − y $ x − y x + (−y)

1 R

C "λ, μ C#

"

#

2 " #

$

1

n $% n+1

n

1

3

$% 4

5 R

x R

0 x 0 x

$ $ &

#

$$ $

6

ρ(x, y) x − y .

7%

% " 4 %

# 8

R ρ(x, 0) %

x R 6 %

x = {ξi}∞i=1, ξi R,

x = {ξi}∞i=1 y = {ηi}∞i=1 λ R

x + y {ξi + ηi}∞i=1, λx {λξi}∞i=1,

R C

x = |x|.

Rn §

!

" E Rn # $ #

" "

$%

(x + y)(t) x(t) + y(t) t E,

(λx)(t) λx(t) t E.

C([a, b]) & !

% [a, b]

x = x C([a,b]) max |x(t)|.

a t b

' () * + !

,%

% $ $-*! - % * !

" . # !

R $ $% /# + $ % "

ε > 0 ε x0 R !

R % "

Uε(x0) {x : x R, x − x0 < ε}.

0 x0 % + ε &

1 " E R %

M> 02 E UM (0)

0 a R % "

E R *$ ε! a " $!

# " E

" E " "

" " " E

3$4 " E R " !

" E % !

" E $% E

" E R

% " E # % * E 1 " E R % !

" E = E

5 E " E R %

" % " R = Rn

*$# $4 #

% " % " !

% " R = Rn

0 x % " ER - Uε(x) + !

"- E

R = Rn!

" E

R \ E

R

!

$ x0 %

ε > 0 nε N : xn Uε(x0) ( xn − x0 < ε) n nε.

,

{xk}∞k=1 x0

* x0

,

% -

+ a R - -

E R )

{xk}∞k=1 xk E xk = a k N ) a

!

. {xk}∞k=1

R

ε > 0 nε N : xk − xj < ε j, k nε.

/ )

(-

0 R

(

)- R

R

1 - 2 & ! -

R Rn 3 4

, 35 3 3 35 6 3

C([a, b]) 6

.

. A B R A

B A B

$ 47 6 6 /-& ! (

) +

C([a, b])

8 R

9% :

) !

. R ;

. R

R

1◦ R R

R R

R

ρ(x, y)

! "

R #! "$ %

%

&

' {xk}∞k=1 %

( ) ( R & %

{xk} {yk} *

ρ({xk}, {yk}) lim ρ(xk, yk).

k→∞

+ , ( '

, {ρ(xk, yk)}∞k=1 R & {xk} %

x0 R *

ρ({xk}, x0) = lim ρ(xk, x0).

k→∞

- ) ( R ' %

{xk} {yk} *

ρ({xk}, {yk}) = 0 . ' %

) ( R %

) , / , ,

* ! " &

! " %

% %

( ) ) , 0 %

* ! "

x0 R

1 R )

! )" R *

* %

R

) %

%

& ) %

) #$ ) *

(

CL([a, b]) = CL1([a, b]) 2 , %

) [a, b] ' 9 , ,

! b

x = x L([a,b]) |x(t)| dt.

a

CL2([a, b])

[a, b]

&! b

! " # $# # %

# !

" (a, b) (−∞, ∞)! & f '

(a, b) → R (a, b) f = 0

[α, β] (a, b)!

" p {1, 2} C0Lp((a, b))

(a, b)

$

(a, b) (−∞, ∞) x' (a, b) → R ! !

# ( "ab |x(t)| dt

$ $

) %# (a, b)

# % ( $ * ! # +!,!-! "

RL1((a, b)) ! ! θ = "ab |θ(t)| dt = 0 # θ =

= 0 * %#

-!

# θ = 0

# θ (

.!

/! 0 $ #

# ( # )

'

◦ θ' [α, β] → R

) [α, β] # # #

% 1 $

! ! % $ $2 # 1

#

# 3

◦ # ) [α, β] θ

"αβ |θ(t)| dt = 0 θ(t) = = 0 %# t θ!

4 % RL1((a, b)) %

# 6

RL1((a, b)) %

% ( *-!

4 x, y RL1((a, b)) 1

"ab |x(t) − y(t)| dt = 0! 5 $

RL1((a, b)) $

! #

6 7 # # '

$ . %

$ !

# # ( $ % %

1 z˜ # % x + y

λx˜ x˜ λ R

# % λx! 9

# $ # %

# '

, # (

- !

6

RL2((a, b)) RL2((a, b)) &

#$ x y

)

+

!b

|x(t) − y(t)|2dt = 0.

a

. # λ R

6

RL2([a, b]) -

§ CL1 CL2 RL1 RL2 L1 L2

θ RL2((a, b)) "ab |θ(t)|2dt = = 0

CLp([a, b]) C0Lp((a, b)) p = 1, 2

/01 (

CL1([−1, 1]) ' ! !

[−1, 1]

. CL1([−1, 1]) &

& ( !

CL1([−1, 1]) 2

# ! # ϕ .

[−1, 1] CL1([−1, 1])

! 1

|ϕ(t) − fk(t)| dt → 0 k → ∞.

−1

3 k → ∞

!0

|ϕ(t)| dt = 0 ϕ(t) = 0 − 1 t 0.

−1

ϕ(t) = 1 0 < δ t 1 δ (0, 1).

ϕ t = 0

CL1([−1, 1])

{fk}∞

k=1

CL1([a, b])

C0((a, b))

(a, b) RL1((a, b))

RL2((a, b))

! "

#$ %& ' % ( " f RL2((a, b)) ε > > 0 ) fε RL2((a, b))

fε = 0 [A, B] (a, b) fε

* [A, B]

f − fε L2((a,b)) < ε.

+ fε

%& ' ,

" M sup |fε| - %& ' %

(a,b)

ϕ C0((a, b))

56 ) 7 / 8 3 %12,9 / #

#

: # ( ;

{fk}∞k=1 $ RL1((0, 1))

RL1((0, 1))

" (0, 1)

k < Ik (0, 1)

.$ $ μIk < ε2−k 00 < ε < 1 k = 1 7 9

RL1((0, 1)) /

RL1((0, 1))

" <

RL1((a, b)) RL2((a, b)) > <

< $ 0

9

( ? %& ' ,

! $ RL1((a, b)) RL2((a, b)) #<

: # / ; . < $ # <

$ " : # ; @ 8

@

: # : # $ @

/ < < [0, 1]

: # 0 #

[a, b]

f ! " [A, B]

# $

! {x [a, b]% f (x) α} &$

α R

' [A, B] k ( A =

= y0 < y1 < . . . < yk = B

f [a, b]

- )*

. / f ! ej $

yj f

ej 0 !

1

k

f (ξi)(xi − xi−1), ξi [xi−1, xi],

i=1

f [xi−1, xi]

f (ξi) + f $

, !

f ) f $

[a, b]

0 " ! f (ξi)

f [xi−1, xi]

2 ! ) $

1

& 3 1 &

4 ( f % [0, 1] → R

1, x ,

f (x) =

0, x

& ) &

1 ,

& 5 1

6 L((a, b)) = L1((a, b)) $

( &(a, b) (−∞, +∞) )7 $

& , !

L1((a, b))

RL1((a, b)) 7 C0L1((a, b))

4 $

L1((a, b))

RL1((a, b)) C0L1((a, b))

6 L2((a, b)) $

( & (a, b)(−∞, +∞) (

& ' )8

& 9 ! $

L2((a, b))

RL2((a, b)) ) 7 C0L2((a, b)) 4 $

L2((a, b))

RL2((a, b)) C0L2((a, b))

L1((a, b)) L2((a, b)) $

( !$

" ! 5

!

!

" # $" %

$

" 6 6

RL1 RL2 & ' (#

§

"

" R &

$ (x, y) % "

,◦ (x1 + x2, y) = (x1, y) + (x2, y) '◦ (λx, y) = λ(x, y) λ R

-" "

& &

.

. x &

/

& )

|(x, y)| x · y . +#

0 x > 0 " %

(tx + y, tx + y) = (x, x)t2 + 2(x, y)t + (y, y) =

= x 2t2 + 2(x, y)t + y 2.

1 $ " 1◦

& # 4(x, y)2−4 x 2· y 2 0,

+#

x 2 + 2 x · y + y 2 = ( x + y )2,

)

x + y x + y .

%

-" n

Rn 1 & "

Rn x = (x1, . . . , xn)

*2# &

"

n

CL2([a, b]) 3 "

& [a, b] $" &

(f, g) "ab f (t)g(t) dt f, g) [a, b] → R

a

- " $ (f, g) "

& & "

$ (f, g) &