Re: [分析] 一個均勻連續的定理(多維)
※ 引述《cxcxvv (delta)》之銘言：
: x∈R^n, E⊂R^n, E is convex
: f is defined on E
: |grad(f)| is bounded
: then f is uniformly continuous on E
: 定理好像是這樣 我不太確定
: 請問哪本書裡有這個證明呢? 謝謝
: (我翻了一下RUDIN沒翻到 APOSTOL有嗎?)
看到convex就想到MVT@@"
Given x,y∈E
f(x)-f(y) = Df(e)(x-y) , e€x-y (x-y代表x與y的連線)
同取euclidean norm
║f(x)-f(y)║=║Df(e)(x-y)║
<= ║Df(e)║║x-y║ , where ║Df(e)║ is the supnorm
<= │Df(e)│║x-y║ , where │Df(e)│ is the 2-norm (見P.S.)
<= M ║x-y║ , │Df(e)│= |▽f(e)| is bdd.
Since g(x)=x is uniformly conti on R^n
so......blabla done.
P.S.
if A€M_mxn(R) (A是mxn的矩陣)
then ║A║<= │A│
where ║A║= sup ║Av║
v€R^n,║v║=1
│A│= √ Σ Σ (a_i_j)^2
i=1~m j=1~n
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 111.251.227.33
推
12/25 01:13, , 1F
12/25 01:13, 1F
討論串 (同標題文章)