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|{k 2 a }−{k 1 a }| = 1−{(k 2 −k 1 )a} N ! ε > 0 ! n ∈ N + {na} < ε 1 − {na} < ε + 1−{na} < ε * 1−{na} = δ δ ∈ (0, ε) 1−{na} = δ na ! na = [na] + 1 − δ (14.6) * ! a > 0 , [na] ≥ 0 * 1−2δ > 0 2na = 2[na]+2−δ 33#b75 5 {2na} = 1 − 2δ 1 − mδ > 0 {mna} = 1−mδ 2 ! m & 1 − (m + 1)δ < 0 , δ > 1 − mδ > 0 {mna} = 1−mδ < δ + \ ε > 0 ! n ε ∈ N + {n ε a } < ε + [ x = 0 x n = {na} " ε > 0 ! n ∈ N + 0 < {na} = δ < ε < 0, 5 + , ! m & 0 < mδ < 1 (m + 1)δ > 1 , {mna} = mδ {(m + 1)na} = δ 1 < δ, ε \ (0, ε) x n = {na} " ! ( x = 0 " [ ε ! ! , n ∈ N + {na} = δ < ε , δ, 2δ, 3δ, . . . , mδ ∈ (0, 1) (14.7) (m + 1)δ > 1 ' δ * {na} = na − [na] = δ ! a = δ + [na] n a 2 ! m ∈ N mδ = 1. 3#b85 x ∈ [0, 1] ! m & |{mna} − x| = |mδ − x| ≤ δ + _ x ∈ [0, 1] ε > 0 ! (x − ε, x + ε) x n = {na} " ! \ [0, 1] ! x n = {na} " % \ & ! M & ( 3W`" 5 M & ! M = M 2 X \M = X\M 1 & ! (; M ! F r M = M ∩ (X\M) = M ∩ (X\M) = ∅ () * +, * *+- + # - & c ' 1 & " \ R n . C [a, b] & ρ (x, y) = $ b a sign |x (t) − y (t) | dt (C[a, b], ρ) c / C (1) [a, b]− g ! & ρ (x, y) = max a ≤t≤b | x (t) − y (t) | (C (1) [a, b], ρ) (C (1) [a, b], ρ) & f : R → R ρ (x, y) = | f (x) − f (y) | R & - 5 4 5 (R, ρ) c * ρ (x, y) = |arctg x − arctg y| (R, ρ) " & Φ − ! ; " ( ! x = (x 1 , x 2 , . . . , x n , 0, 0, . . .) " & * ρ 1 (x, y) = ∞ i =1 | x i − y i | ; ρ 2 (x, y) = max 1≤i<∞ | x i − y i | (Φ, ρ 1 ) (Φ, ρ 2 ) & X = (−π, π) & ρ (x, y) = ## ##sin x − y 2 ## ## (X, ρ) & $ P f ! ^ & & x, y ∈ P ! 5 ρ 1 (x, y) = max 0≤t≤1 | x (t) − y (t) | + max 0≤t≤1 | x (t) − y (t) | ; 5 ρ 2 (x, y) = max 0≤t≤1 | x (t) − y (t) | + max 0≤t≤1 | x (t) − y (t) | ; !5 ρ 3 (x, y) = max −1≤t≤1 | x (t) − y (t) | + | x (0) − y (0) | (P, ρ 1 ), (P, ρ 2 ), (P, ρ 3 ) " & % m 2 n : m, n ∈ Z & R ! & Z, 1 n : n ∈ Z, n = 0 1 n + 1 m : n, m ∈ Z, n · m = 0 & " R ! ! ' ! ! & = & 3$#$"$%`5 . = & ! / = & + = & ! + ( 0 K = ∅ = & [0, 1] ! ! = & & ( K = K R A & A, 0 A, 0 A, 0 A, 0 A, 0 0 A & ( ! $ A ⊂ (X, ρ) ! ! X \A & ! % * A ! ! & X \A & ! ! & 2 A & A X \A & ! ' Φ − ; " & c 0 p (p ≥ 1) ! + c m ! . * A & B B C ! + A & C ! / P − ! & & C [a, b] ! [a, b] a = t 1 < t 2 < · · · < t n = b x 1 , x 2 , . . . , x n , x (t i ) = x i , i = 1, n [t i , t i +1 ] ! x (t) '! ! & C [a, b] ! '! 3 ! & & 5 & L 1 [a, b] ! " 2 & L p [a, b] (p ≥ 1) ! " [a, b] a = t 1 < t 2 < · · · < t n = b (t i , t i +1 ), i = 1, n − 1 t i " Z & " $ L p [a, b] (p ≥ 1) ! " & ! + % L p [a, b] (p ≥ 1) ! & '! L p [a, b] (p ≥ 1) & " & ! \ C [a, b] & L p [a, b] ! ' [a, b] ! " L p [a, b] & ( x (t) ε > 0 ! p (t) & + ρ (x, p) = $ b a | x (t) − p (t) | p dt 1/p < ε . . #b`W" p (t) & ! ; " / 2 & & c 2 & G ! & ! ! - G = ∪ n B (x n , r n ) , B (x i , r i ) ∩ B (x j , r j ) = ∅, i = j 2 & F & & M ! N & \ & & M N #" & , M ∪ N #" & $ \ + n & P ≤n & C [a, b] ! ! % P = ∞ n =1 P ≤n ! & & C [a, b] #" " & & L 2 [a, b] & L 1 [a, b] #" & " ' x n (t) − t ∈ R ! lim n →∞ x n (t) = x 0 (t) x 0 (t) & #" & . C [a, b] & ρ (x, y) = $ b a sign | x (t) − y (t) | dt (C[a, b], ρ) & $/ C [a, b] M n = {x : |x(t ) − x(t )| ≤ n · |t − t | , ∀t , t ∈ [a, b]} & & ! ! $ C [a, b] /& ! & " M = ∞ n =1 M n 3#ba>" 5 #" & M & & C [a, b] ! $ C [a, b] n ∈ N D n = {x : x ( Download 1.57 Mb. Do'stlaringiz bilan baham: 
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