Re: [分析] 請教一個問題

[keroro321 (日夕)]

※ 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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