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t) ∈ C[a, b]
max a ≤t≤b |x (t)| ≤ n } & & ! ! $ C [a, b] g ! & C (1) [a, b] = ∞ n =1 D n 3 D n − #ba%" 5 #" & & ! $ ! ! & & ! ! $$ d ! ! & ! ! $% (X, ρ) M ⊂ X #" & , X \M & X ! $& (X, ρ) G n ⊂ X, n ∈ N ! ! & , M = ∞ n =1 G n & ! $' M & ! ! ! 0 ¯ M = ∅ +" $. X M ⊂ X #" & , X \M %" & %/ * y 0 ∈ B(x 0 , r ) B (y 0 , r ) ⊂ B(x 0 , 2r) " % ' A & ε − ; V ε (A) = x ∈ X : inf y ∈A ρ (x, y) < ε , ¯ A = ε> 0 V ε (A) % 1 ! [a, b], (a, b), Z, Q, [a, ∞), ∅ & ! " & % ! ! & & ! ! % F r (A ∪ B) ⊂ F rA ∪ F r B * A ∩ B = ∅ F r (A ∪ B) = F rA ∪ F r B %$ 2 & & ! " & %% {x n } ⊂ [a, b] (α, β) ⊂ [a, b] ! n (α; β) x 1 , x 2 , . . . , x n (α, β) * (α, β) ⊂ [a, b] ! lim n →∞ n (α; β) n = β − α b − a + {x n } " [a, b] , 0, 1, 1 2 , 1 3 , 2 3 , 1 4 , 2 4 , 3 4 , . . . , 1 k , 2 k , . . . , k − 1 k , . . . " " [0, 1] c %& α − {n α} = n α − [n α], ( n α " [0, 1] %' #b78" 1, 5, 5 2 , . . . , 5 n , . . . " 5 n " 3 5 #` & %. X {f n } X * x ∈ X ! ! lim n →∞ f n (x) = f(x) + f & #" & &/ * f : R → R ! f (x) & %" &" $9 (+0+ 0- X = (X, ρ) Y = (Y, d) f f X Y " * ε > 0 ! δ > 0 + ρ (x, x 0 ) < δ ! ! x ∈ X ! d (f(x), f(x 0 )) < ε f x 0 ∈ X * f X f X * " ε > 0 ! δ > 0 + ρ (x, y) < δ ! ! x, y ∈ X ! d (f(x), f(y)) < ε " + f X * f : X → Y f f −1 f - X Y * (X, ρ) (Y, d) " ! f x 1 , x 2 ∈ X ! ρ (x 1 , x 2 ) = d (f (x 1 ) , f (x 2 )) f " X Y " $ f : R 2 → R x y ! ! & y x ! ! & f (x, y) ! & " 2 f (x, y) ! f (x, y) = a 0 (x) + a 1 (x) y + a 2 (x) y 2 + · · · + a n (x) y n , (15.1) f (x, y) = b 0 (y) + b 1 (y) x + b 2 (y) x 2 + · · · + b m (y) x m . (15.2) ' f x y ! ! g ! ! * y = 0 3#a#5 3#a%5 f (x, 0) = a 0 (x) = b 0 (0) + b 1 (0) x + b 2 (0) x 2 + · · · + b m (0) x m . 3#a#5 3#a%5 y ! y = 0 a 1 (x) = b 0 (0) + b 1 (0) x + b 2 (0) x 2 + · · · + b m (0) x m f (x, y) 3#a#5 3#a%5 y ! n − y = 0 " a n (x) = b (n) 0 (0) + b (n) 1 (0) x + b (n) 2 (0) x 2 + · · · + b (n) m (0) x m a 0 (x), a 1 (x), . . . , a n (x) ! & " 3#a#5 f (x, y) ! & ! $ 2 f : R → R G − ! F − & & " f (G) & ! f (F ) & & ! "# * F ! & 3 & 5 & f (F ) & & ! & \ F & ! & f (x) = ⎧ ⎨ ⎩ cos x, agar x ∈ [−4π, 4π], 1 + (x − 4π) 2 : x 2 , agar |x| > 4π. G = (−2π − 1, 1) ! & f (G) = [−1, 1] & F = [4π, ∞) & & f (F ) = [1, 2) & & $ f : C[0, 1] → L 1 [0, 1] f (x(t)) = x(t) x 0 ∈ C[0, 1] ρ (f(x), f(x 0 )) = 1 $ 0 |x(t) − x 0 (t)| dt ≤ max 0≤t≤1 |x(t) − x 0 (t)| 1 $ 0 dt = ρ(x, x 0 ) ' ε > 0 ! δ = ε ρ (x, x 0 ) < δ ! ! x ∈ X ρ (f(x), f(x 0 )) < ε x 0 ∈ C[0, 1] ! f " $ f : C[0, 1] → C[0, 1] 5 f (x(t)) = t + 0 x (s)ds 4 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)) = x(t α ), α ≥ 0 , " c ! "# 5 5 5 !5 ' 5 ! f '! x, y ∈ C[0, 1] ! ρ (f(x), f(y)) = max 0≤ t ≤1 ## ## $ t 0 (x(s) − y(s)) ds ## ## ≤ max 0≤ s ≤1 |x(s) − y(s)| $ t 0 ds, ( ρ (f(x), f(y)) ≤ ρ(x, y) 1 (; δ ε ρ (x, y) < δ ! ! x, y ∈ X ρ (f(x), f(y)) < ε + \ f " 1 $$ R ρ 1 (x, y) = |x − y| ρ 2 (x, y) = ⎧ ⎨ ⎩ 1, agar x = y 0, agar x = y 1 ( ρ 1 ρ 2 , C 1 > 0, C 2 > 0 + ! x = y, x, y ∈ (−∞, ∞) ! C 1 ρ 1 (x, y) ≤ ρ 2 (x, y) ≤ C 2 ρ 1 (x, y) ⇐⇒ C 1 |x − y| ≤ 1 ≤ C 2 |x − y| + / |x − y| = 1 2C 2 + \ ρ 1 ρ 2 $% C [a, b] & ρ ∞ (x, y) = max a ≤t≤b |x(t) − y(t)|, ρ 2 (x, y) = . b + a |x(t) − y(t)| 2 dt , ρ 1 (x, y) = b + a |x(t) − y(t)|dt, ρ 4 (x, y) = b + a sign|x(t) − y(t)| dt ' ρ 1 ρ ∞ C [a, b] y (t) = 0 x n (t) = ⎧ ⎨ ⎩ 1 − n(t − a), agar t ∈ [a, a + 1 n ] 0, agar t ∈ (a + 1 n , b ] ! ρ ∞ (x n , y ) = 1 ≤ Cρ 1 (x n , y ) ! C > 0 + d n → ∞ ρ 1 (x n , y ) = a + 1 n $ a (1 − n(t − a))dt = 1 n − n 2n 2 = 1 2n \ Download 1.57 Mb. Do'stlaringiz bilan baham: 
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