※ 引述《Eliphalet (真係廢到冇朋友)》之銘言： : 有一個問題想請教大家 : 令 D 是 R^n 上的一個非空的開集合, f_k : D -> R 是 differentiable 的函數 : 假設 f_k -> f pointwise 且 ▽f_k -> g uniformly on D : 那麼會不會有以下的結果 : f 是 differntiable ? : ▽f_k -> ▽f uniformly on D ? : 我知道 n=1 時是對的 那麼當 n >= 2 時呢 ? 答案是肯定的 ! f_k need not to be of class C^1. 參考的書來自 (Rudin , Principles Of Mathematical Analysis ) 稍加改一下 n = 1 的證明 , 就可得到 n ≧ 2 情形. 只證open ball的情形就好了,在證明之前,先提一下下面定理7.11 . Theorem 7.11: Suppose φ_n —> φ uniformly on a set E in a metric space . Let x be a limit point of E , and suppose that lim φ_n(t) = An (n =1,2,3.........) t->x Then {An} converges , and lim φ(t) = lim An t->x n->∞ Theorem 7.17: ( 0<r<∞ ) {fn:Br—>|R} is a sequence of functions (Br is an open ball in |R^n), differentiable on Br and such that {fn(x_0)} converges for some point x_0 on Br , If {Dfn} converges uniformly on Br , then {fn} converges uniformly on Br ,to a function f , and Df(x) = lim Dfn(x) (for all x in Br) n->∞ Proof: ( Notation: x,t are elements of Br , Dfn(x) = Ln(x) , Ln(x)－>L(x) ) Let ε>0 be given . Choose N such that n≧N , m≧N ,implies ∣fn(x_0)-fm(x_0)∣< (ε/2) and ∥Dfn(t)-Dfm(t)∥< (ε/(4r)) (for all t in Br) ∥ ∥ Ln(t) Lm(t) we apply the mean value theorm (|R^n) to fn(t)-fm(t) ∣fn(t)-fm(t)-(fn(x)-fm(x))∣≦∣(Ln(c)- Lm(c))‧(t-x)∣ ≦ ∥Ln(c)- Lm(c)∥∥t-x∥< (ε/(4r))∥t-x∥-------------(1) where c is a ponit on the line segment joining x and t . ∴∣fn(x)-fm(x)∣≦∣fn(x)-fm(x)-(fn(x_0)-fm(x_0))∣+∣fn(x_0)-fm(x_0)∣ < (ε/(4r))*(2r)+(ε/2) =ε { fn(x) } converges uniformly on Br. Let f be the limit function of { fn(x) } . Now , fix a point x in Br . ∣fn(t)-fn(x)-Ln(x)‧(t-x)∣ Let φ_n(t) = ──────────────── ∥t-x∥ ∣f(t)-f(x)-L(x)‧(t-x)∣ Let φ(t) = ──────────────── ∥t-x∥ ∥φ_n(t) - φ_m(t)∥ ≦ ∣fn(t)-fn(x)-Ln(x)‧(t-x)-(fm(t)-fm(x)-Lm(x)‧(t-x))∣ ──────────────────────────── (..from (1)) ∥t-x∥ ＜ ε/(4r)+ε/(4r) =ε/(2r) {φ_n} converges uniformly on Br -∣ ∣=>φ_n－>φ uniformly on Br lim φ_n(t) = φ(t)----------------∣ n->∞ lim φ_n(t) = 0 = An t->x By Theorem 7.11 lim φ(t) = lim An = 0 t->x n->∞ that is Df(x) = lim Dfn(x) n->∞ Q.E.D 有打錯麻煩提醒我 ! --

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