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funksional analiz misol va masalalar yechish 1 qism
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(M) & ! * " & X * f : X → R ! " X & \ X ! + X & & * M − & & F − & & M ∩ F = ∅ d (M, F ) = inf x ∈M,y∈F ρ (x, y) > 0. & 2 M F & & M ∩ F = ∅ d (M, F ) = inf x ∈M,y∈F ρ (x, y) = 0 & $ , 5 { t α } , α > 0; 5 {sin α t} , α ∈ R; !5 1 α + t 2 , α > 0; 5 t α 1 + t 2 , α > 0; 5 {ln α t } , α > 0 [0, 1] + c < ! c & % K & C (K) & ! 3 & 5 & * ρ (x, y) = max t ∈K |x(t) − y(t)| C (K) & & & X K ⊂ X & & x ∈ X ! y ∈ K + ρ (x, y) = inf z ∈K ρ (x, z) ( x ! K + & ' X K & * ε > 0 ! K & & ε + K & " & . * C (1) [a, b] K & & & & / C [0, 1] M 1 = {x ∈ C[0, 1] : |x(t)| ≤ 1} ; M 2 = {x ∈ C[0, 1] : |x(t)| ≤ 1, |x (t)| ≤ 2} ; M 3 = {x ∈ C[0, 1] : |x(t)| ≤ 1, |x (t)| ≤ 2, |x (t)| ≤ 3} ; M 4 = {x ∈ C[0, 1] : |x(t)| ≤ 1, |x (t)| ≤ 2} ; M 5 = {x ∈ C[0, 1] : |x (t)| ≤ 1, |x (t)| ≤ 2} , & & & c & K = [0, 1] × [0, 1] g ! ## ## ∂f ∂t 1 ## ## ≤ 1, ## ## ∂f ∂t 2 ## ## ≤ 1; f(0,0) = 1 ! f (t 1 , t 2 ) & C (K) & & {a n } M = {x ∈ 2 : |x n | ≤ a n } )& &" & ) 2 & c & K ⊂ X & f n & * x ∈ K ! {f n (x) } ! f 1 (x) ≤ f 2 (x) ≤ · · · ≤ f n (x) ≤ · · · lim n →∞ f n (x) = f(x) {f n } " " f ( C (K) ρ (f n , f ) → 0. - 0 # - " * R *5 ρ (x, y) = |x − y| '5 ρ (x, y) = |x| − |y| d5 ρ (x, y) = |x − y| 2 \5 ρ (x, y) = |x| + |y| R n *5 ρ (x, y) = . n k =1 (x k − y k ) 2 '5 ρ (x, y) = n k =1 |x k − y k | d5 ρ (x, y) = max 1≤k≤n |x k − y k | \5 ρ (x, y) = n k =1 (x k − y k ) 2 R n 1 *5 ρ (x, y) = . n k =1 (x k − y k ) 2 '5 ρ (x, y) = n k =1 |x k − y k | d5 ρ (x, y) = max 1≤k≤n |x k − y k | \5 ρ (x, y) = n k =1 (x k − y k ) 2 R n ∞ *5 ρ (x, y) = . n k =1 (x k − y k ) 2 '5 ρ (x, y) = n k =1 |x k − y k | d5 ρ (x, y) = max 1≤k≤n |x k − y k | \5 ρ (x, y) = n k =1 (x k − y k ) 2 $ C [a, b] *5 ρ (x, y) = . b + a |x(t) − y(t)| 2 dt '5 ρ (x, y) = b + a |x(t) − y(t)| dt d5 ρ (x, y) = max a ≤t≤b |x(t) − y(t)| \5 ρ (x, y) = b + a |x(t) − y(t)| 2 dt % C 1 [a, b] *5 ρ (x, y) = . b + a |x(t) − y(t)| 2 dt '5 ρ (x, y) = b + a |x(t) − y(t)| dt d5 ρ (x, y) = max a ≤t≤b |x(t) − y(t)| \5 ρ (x, y) = b + a |x(t) − y(t)| 2 dt & C 2 [a, b] *5 ρ (x, y) = . b + a |x(t) − y(t)| 2 dt '5 ρ (x, y) = b + a |x(t) − y(t)| dt d5 ρ (x, y) = max a ≤t≤b |x(t) − y(t)| \5 ρ (x, y) = b + a |x(t) − y(t)| 2 dt ' p , p ≥ 1 *5 ρ (x, y) = sup 1≤k≤∞ |x k − y k | '5 ρ (x, y) = ∞ k =1 e −k |x k − y k | d5 ρ (x, y) = p . ∞ k =1 |x k − y k | p \5 ρ (x, y) = ∞ k =1 |x k − y k | p . 2 *5 ρ (x, y) = sup 1≤k≤∞ |x k − y k | '5 ρ (x, y) = ∞ k =1 e −k |x k − y k | d5 ρ (x, y) = . ∞ k =1 |x k − y k | 2 \5 ρ (x, y) = ∞ k =1 (x k − y k ) 2 / Z & Q ! & *5 R '5 Q d5 ∅ \5 R\Q Z & Q ! & *5 R '5 Q d5 ∅ \5 R\Q Z & Q ! & *5 R '5 Q d5 ∅ \5 R\Q ' & Z ! & & *5 R '5 Q d5 ∅ \5 Z ' & Z ! & & *5 R '5 Q d5 ∅ \5 Z $ ' & Z ! & & *5 R '5 Q d5 ∅ \5 Z % R ! & & *5 (0, 2) '5 (0, 2] d5 [0, 2) \5 [0, 2] & R & & & *5 (0, 1) '5 [0, 1) d5 (0, 2] \5 [0, 4] ' R ! & & *5 [0, 1] '5 (−∞, 0) d5 Q \5 (0, ∞) . (X, ρ) ! & (; *5 F ⊂ X & X 4 '5 F ⊂ X " & & 4 d5 F ⊂ X " ! & 4 \5 F ⊂ X " ! / < ! + 1) ρ(x, y) = 0 ⇔ x = y; 2) ρ(x, y) = ρ(y, x), ∀x, y ∈ X 3) ρ(λx, y) = λρ(y, x), 4) ρ(x, y) ≤ ρ(x, z) + ρ(z, y), ∀x, y, z ∈ X. *5 # % ` '5 # % b d5 # ` b \5 # % ` b < (X, ρ) & & (; c # * F & ! 4 % * F ! ! 4 ` * F = [F ] 4 b * F ! ! & *5 # % ` '5 # ` b d5 % ` b \5 # % ` b < (X, ρ) ! & (; c # * F & ! 4 % * F ! ! 4 ` * F = [F ] 4 b * F ! & & *5 # % ` '5 # ` b d5 % ` b \5 % b R ! & & *5 Q " & '5 & d5 Z " & \5 N " & R ! & & *5 & '5 R\Q " & d5 Z " & \5 N " & $ R ! ! & & *5 Q " & '5 Z " & d5 R\Q " & \5 [0, ∞) % < + #5 .! & ! & & %5 .! & ! ! & `5 2 ! & ! & b5 2 ! & ! & *5 # % b '5 # % ` d5 # b \5 # ` & < + #5 _ & & ! ! & %5 _ & & ! & & `5 2 & & & & b5 2 & & & & *5 # % b '5 # % ` d5 # b \5 # ` ' < c #5 R %5 R n `5 C [a, b] b5 2 a5 C 2 [a, b]. *5 # % ` b '5 # % b a d5 # % ` a \5 % ` b a . C 1 [−1, 1] " " c 1) f n (x) = x n , 2) f n (x) = 1 − x n , 3) f n (x) = (sin x) n , 4) f n (x) = (cos x) n *5 # ` b '5 # % d5 % b \5 # % ` / C 2 [0, 1] x n (t) = t n + t n +1 " & *5 x (t) = 0 '5 x (t) = 2t d5 x (t) = 1 \5 x (t) = 2 R n = "' *5 n k =1 |a k · b k | ≤ n k =1 a 2 k · n k =1 b 2 k '5 n k =1 |a k · b k | ≤ n k =1 |a k | p 1 p · n k =1 |b k | q 1 Download 1.57 Mb. Do'stlaringiz bilan baham: 
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