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× X X " & 7, ! ρ : X × X → R X " .& ρ (x, y) ≥ 0, ∀ x, y ∈ X, ρ(x, y) = 0 ⇔ x = y $ & /& ρ (x, y) = ρ(y, x), ∀ x, y ∈ X $ & 0& ρ (x, z) ≤ ρ(x, y) + ρ(y, z) ∀ x, y, z ∈ X $ & (X, ρ) . ( (X, ρ) + X ; " * X & ρ 1 , ρ 2 , . . . , ρ n (X, ρ 1 ), (X, ρ 2 ), . . . , (X, ρ n ) X 1 , X 2 , . . . , X n : 7, ! C 1 > 0 C 2 > 0 x, y ∈ X C 1 ρ 1 (x, y) ≤ ρ 2 (x, y) ≤ C 2 ρ 1 (x, y) ρ 1 ρ 2 ' " R n = {x = (x 1 , x 2 , . . . , x n ) , x i ∈ R} − n ! m − " ! ! " & c − ! ! " & c 0 − ! ! " & 2 − + ! ! " & ( 2 = x = (x 1 , x 2 , . . . , x n , . . . ) : ∞ n =1 |x n | 2 < ∞ . p (p ≥ 1) − p − + " ! ! " & ( p = x = (x 1 , x 2 , . . . , x n , . . . ) : ∞ n =1 |x n | p < ∞ . p = 1 p & f ! ! " C [a, b]− [a, b] ! & M [a, b] − [a, b] ! & C (1) [a, b] − [a, b] ! g ! & C (n) [a, b]−[a, b] n g " ! & V [a, b] − [a, b] ! & AC [a, b] − [a, b] ! & L p [a, b]− ! [a, b] (p ≥ 1) p − + / ( ! & L (0) p [a, b] ⊂ L p [a, b] & * f − g ∈ L (0) p [a, b] ϕ " ' L p [a, b] : + : & L p [a, b] p = 1 [a, b] / ( ! : & L 1 [a, b] p = 2 [a, b] " / ( ! : L 2 [a, b] ' & R 2 & x = (x 1 , x 2 ) y = (y 1 , y 2 ) ! ρ 1 (x, y) = | x 1 − y 1 | + | x 2 − y 2 | ; ρ 2 (x, y) = | x 1 − x 2 | + | y 1 − y 2 | ; ρ 3 (x, y) = | x 1 − y 2 | + | x 2 − y 1 | ; ρ 4 (x, y) = | x 1 x 2 − y 1 y 2 | c ! "# ρ 1 " ρ 1 (x, y) ≥ 0 ; ! ρ 1 (x, y) = |x 1 −y 1 |+|x 2 −y 2 | = 0 , |x 1 −y 1 | = |x 2 −y 2 | = 0 ' x 1 = y 1 , x 2 = y 2 ( x = y. [ x = y ( x 1 = y 1 , x 2 = y 2 ' ρ 1 (x, y) = |x 1 − y 1 | + |x 2 − y 2 | = 0 \ #" + < ρ 1 (x, y) = |x 1 − y 1 | + |x 2 − y 2 | = |y 1 − x 1 | + |y 2 − x 2 | = ρ 1 (y, x) %" + ! 0 ρ 1 (x, z) = |x 1 − z 1 | + |x 2 − z 2 | = |x 1 − y 1 + y 1 − z 1 | + |x 2 − y 2 + y 2 − z 2 | ≤ ≤ |x 1 − y 1 | + |y 1 − z 1 | + |x 2 − y 2 | + |y 2 − z 2 | = ρ 1 (x, y) + ρ 1 (y, z) `" + ! 2 R 2 , ρ 1 = R 2 1 ρ (x, y) = sup 1≤n<∞ |x n − y n | x, y ∈ c 0 " ! "# {x n } {y n } " ! ! ! 2 ! sup 1≤n<∞ |x n − y n | ! x, y ∈ c 0 ! n ∈ N |x n − y n | ≥ 0 ρ (x, y) = sup n ≥1 |x n − y n | ≥ 0 ! [ ρ (x, y) = sup n ≥1 |x n − y n | = 0 ! n ∈ N |x n − y n | = 0 ' x = y * x = y ρ (x, y) = 0 \ #" + |x n − y n | = |y n − x n | %" + ! ,! ! |x n − z n | ≤ |x n − y n | + |y n − z n | , sup n (x n + y n ) ≤ sup n x n + sup n y n ! \ ρ : c 0 × c 0 → R ! ! + ρ (x, y) = (x − y) 2 , x, y ∈ R ! "# ρ (x, y) = (x − y) 2 , x, y ∈ R 1− 2− ' ! ! ! ' ! x = 0, y = 3, z = 5 , ρ (x, z) = 25, ρ(x, y) = 9, ρ(y, z) = 4 25 = ρ(x, z) > ρ(x, y) + ρ(y, z) = 9 + 4 = 13. \ ρ ! ! ! X = AC[0, π], x (t) = sin t, y (t) = 0 x ∈ X y ∈ X & ! "# AC [0, π] x y ρ (x, y) = |x(0) − y(0)| + V π 0 [x − y] ( x (t) = sin t [0, π] % 2 ! x (t) = sin t y (t) = 0 - ρ (x, y) = |x(0) − y(0)| + V π 0 [x − y] = V π 0 [x] = 2. () * +, * *+- + # - $ R ρ (x, y) = | arctgx − arctgy | " ρ 1 (x, y) = arctg | x − y| R & c % R n & c ρ 1 (x, y) = n k =1 | x k − y k | ; ρ 2 (x, y) = ## ## max 1≤k≤n (x k − y k ) ## ##; ρ 3 (x, y) = ⎧ ⎨ ⎩ 1, agar x = y 0, agar x = y ; ρ 4 (x, y) = n k =1 sign | x k − y k | . & R 3 ! 3 5 & ρ (x, y) = arccos(x, y) = arccos(x 1 y 1 + x 2 y 2 + x 3 y 3 ) ' [0, 1] ! ! & X * μ − / ! ρ (A, B) = μ (A Δ B) X . '! & P & ρ (p, q) = ∞ i =0 ## #p (i) (0) − q (i) (0) ## # / 0 & N ρ 1 (n, m) = |e in − e im | 4 ρ 2 (n, m) = ⎧ ⎨ ⎩ 1 + 1 n + m , agar n = m 0, agar n = m ; ' & Z ρ 1 (n, m) = |m − n | √ 1 + m 2 · √ 1 + n 2 ; ρ 2 (n, m) = 10 −k , 3 k | n − m | n = m k = ∞ 5 , | z | < 1 ! & & " ρ (z 1 , z 2 ) = ln 1 + ## ## z 2 − z 1 1 − z 2 ¯z 1 ## ## 1 − ## ## z 2 − z 1 1 − z 2 ¯z 1 ## ## ' 1 [a, b] x ! α β + ! t 1 , t 2 ∈ [a, b] ! | x (t 1 ) − x (t 2 ) | ≤ β · | t 1 − t 2 | α + x α : ' ! ! &" H α [a, b] * α > 1 H α [a, b] * 0 < α ≤ 1 ρ (x, y) = max a ≤t≤b |x(t) − y(t)| + sup t 1 =t 2 |(x(t 1 ) − y(t 1 )) − (x(t 2 ) − y(t 2 ))| |t 1 − t 2 | α H α [a, b] & H α [a, b] Y α = 1 1 [ Download 1.57 Mb. Do'stlaringiz bilan baham: |
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