推 math1209 :1. 我也覺得用不到. 2. 的確是 AA-thm. 你要作的是 04/19 21:19
→ math1209 :找到 {f_n(x)} 是 equi-continuous 且 "pointwise" 04/19 21:20
→ math1209 :bounded. (我知道有些書會寫 uniformly bdd.) 04/19 21:20
→ smartlwj :謝謝math大 我再想想看第二題 04/19 21:55
1. Consider a real-valued function f on an interval. Suppose that f
is absolutely continuous and differentiable so f' is integrable.
Show that f satisfies a Lipschitz condition if and only if |f'|
is bounded.
2. Let fn:[0,2] → R be a sequence of differentiable functions whose
derivatives are uniformly bounded. Prove that if fn(1) is bounded
as n → ∞, then the sequence {fn} has a sbusequence that converges
uniformly on [0,2].
請問這兩題該怎麼做呢?
第一題 (<=) 這個方向有做出來,用MVT
而(=>) 我是這樣寫 since f is Lip. => |f(x)-f(x')|≦M|x-x'|
=> |f(x)-f(x')| ≦ M
------------
|x-x'|
接下來讓 x → x' 就可以得到 |f'| 有界
可是我的問題在,我似乎沒有用到絕對連續的條件???
是不是哪裡弄錯了呢??
第二題 一開始沒想法,後來想到似乎可以用Ascoli-Arzela thm的樣子
但我還是不知道該怎麼証....
請指點一下 謝謝!!
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