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1 ρ ∞ () * +, * *+- + # - $& f : [0, 1]×[0, 1] → R x y ! y x ! (x 0 , y 0 ) +" f ! $' f (0, 1) ! g ! * x ∈ (0, 1) ! n = n(x) ∈ N + f (n) (x) = 0 f & $. (X, ρ) (Y, d) f : X → Y ! " - 5 G ⊂ Y ! & ! f −1 (G) ⊂ X ! & 4 5 F ⊂ Y & & ! f −1 (F ) ⊂ X & & 4 !5 {x n } ⊂ X ! " ! {f(x n )} ⊂ Y " ! $/ " " ! c $ * X f : X → Y $ f i : X → Y, (i = 1, 2) , M = {x ∈ X : f 1 (x) = f 2 (x)} & & $ f i : X → Y, (i = 1, 2) X ! M & * ! x ∈ M ! f 1 (x) = f 2 (x) f 1 ≡ f 2 ( X $ f : X → Y < & c M ⊂ X & ! - 5 x ∈ ¯ M ⇒ f(x) ∈ f(M) 4 5 x ∈ 0 ¯ M ⇒ f(x) ∈ 0 f (M) 4 !5 x ∈ M ⇒ f(x) ∈ (f(M)) 4 5 x ∈ F r M ⇒ f(x) ∈ F r (f(M)) $$ X & f : X → Y M X ! lim x →a f (x) + lim x →a f (x) = f (a) M & & $% S = {z ∈ C : |z| = 1} ρ (z 1 , z 2 ) = |z 1 − z 2 | f : S → R ! z 0 ∈ S + f (z 0 ) = f(−z 0 ) \ " $& f : C[a, b] → C[0, 1] f (x(t)) = x(a+(b−a)t), 0 ≤ t ≤ 1 ' - 5 5 c $' f (x(t)) = x(t 2 ) 5 f : C[−1, 1] → C[0, 1] 4 5 f : L p [−1, 1] → L p [0, 1] 4 !5 f : C[−1, 1] → L 1 [0, 1] " , $. f (x(t)) = x 2 (t) 5 f : C[0, 1] → C[0, 1] 4 5 f : L p [0, 1] → L 2 [0, 1] 4 !5 f : L 1 [0, 1] → L 2 [0, 1] 4 5 f : L 2 [0, 1] → L 1 [0, 1] 4 5 f : L 1 [0, 1] → L 1 [0, 1] , $ / f : L 2 [0, 1] → L 2 [0, 1] 5 f (x(t)) = t + 0 x (s)ds; 5 f (x(t)) = 1 + 0 sin(t − s)x(s)ds 4 !5 f (x(t)) = t + 0 x 2 (s)ds 4 5 f (x(t)) = t + 0 x (s α )ds, α ≥ 0 , " c $ f : L 2 [0, 1] → L 2 [0, 1] 5 f (x(t)) = u(t) · x(t), u ∈ C[0, 1] 4 5 f (x(t)) = x(t α ), α > 0 , $ R $ X, Y − Y − * M ⊂ X ! f : M → Y F : X → Y + F | M = f, ( x ∈ M ! F (x) = f(x). $ /& ! $ $ K (t, s) [a, b] × [a, b] ! Ax (t) = $ b a K (t, s)x(s)ds A : C[a, b] → C[a, b] /& " $ % . ! K (t, s) ! $ b a $ b a |K(t, s)| 2 dtds ≤ M A : L 2 [a, b] → L 2 [a, b] Ax (t) = $ b a K (t, s)x(s)ds $ & f : X → Y g : Y → Z /& " ! , g ◦ f : X → Z 3/& ! 5 c $ ' [0, 1] /& " $ . (X, ρ) ρ : X × X → R 5 4 5 c $ / (X, ρ) A = ∅, A ⊂ X & d : X → R d (x) = inf y ∈A ρ (x, y) 2 " $ R 2 ρ (x, y) = " (x 1 − y 1 ) 2 + (x 2 − y 2 ) 2 C & " & d (z 1 , z 2 ) = |z 1 − z 2 | ' $ X Y X ×Y Y ×X $ c R × c 0 $ C [0, 1] C [a, b] $ $ $ % R ! & $ & R 2 3 ρ (x, y) = " (x 1 − y 1 ) 2 + (x 2 − y 2 ) 2 5 ! " & $ ' " ! ! && $ . ' X & ρ 1 ρ 2 X & ! ! $ / X ! & $ R n ρ 1 , ρ 2 = ρ ρ ∞ " $ [ ! 3 5 " ! ! 3 5 " $ [ ! 3 & 5 & ! " ! 3 & 5 $ ρ 1 ρ 2 * (X, ρ 1 ) 5 4 5 & 4 !5 (X, ρ 2 ) $ $ C n & ρ ∞ (x, y) = max 1≤i≤n |x i − y i |, ρ 1 (x, y) = n i =1 |x i − y i | ρ 2 (x, y) = . n i =1 |x i − y i | 2 $ % X & [a, b] ! ! 2 & ρ 1 (x, y) = ⎛ ⎝ b $ a |x(t) − y(t)| p 1 dt ⎞ ⎠ 1/p 1 ; ρ 2 (x, y) = ⎛ ⎝ b $ a |x(t) − y(t)| p 2 dt ⎞ ⎠ 1/p 2 p 1 = p 2 , (p 1 ≥ 1, p 2 ≥ 1) $ & Y ; $ ' f : X → Y ; M ⊂ X & , - 5 M ! & f (M) ! 4 5 M & & f (M) & 4 !5 f ( ¯ M ) = f(M) $ . Y & ! &" $$/ R & ρ 1 (x, y) = |x − y| ρ 2 (x, y) = |arctg x − arctg y| (R, ρ 1 ) (R, ρ 2 ) " x n = n " (R, ρ 2 ) " (R, ρ 1 ) (R, ρ 1 ) (R, ρ 2 ) \ ! ? & $$ (X, ρ) * ρ 1 (x, y) = ρ (x, y) 1 + ρ(x, y) (X, ρ) (X, ρ 1 ) $$ R R 2 , n = m R n R m $$ C (2) [a, b] & ρ 1 (x, y) = max a ≤t≤b |x(t) − y(t)| + max a ≤t≤b |x (t) − y (t)| + max a ≤t≤b |x (t) − y (t)| , ρ 2 (x, y) = max a ≤t≤b |x(t) − y(t)| + max a ≤t≤b |x (t) − y (t)| $$ (X, ρ) (Y, d) * (Y, d) (X, ρ) 3 5 (X, ρ) (Y, d) $ & * n ≤ m R n R m + ' ρ (x, y) = 4 5 5 6 k i =1 |x i − y i | 2 , ρ ∞ (x, y) = max 1≤i≤k |x i − y i | , ρ 1 (x, y) = k i =1 |x i − y i | ( Download 1.57 Mb. Do'stlaringiz bilan baham: 
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