Re: [分析] function space
※ 引述《jacky7987 (憶)》之銘言：
: 苦惱許久
: 因為我一直突破不了積分QAQ
: Let H be a metric space composed of continuous functions on [0,1] with metric
: d defined by
: 1
: d(f,g)=∫|f(x)-g(x)|dx
: 0
: |f(x)-f(y)|
: Let ∥f(x)∥ = max |f(x)| and ∥f(x)∥= ∥f(x)∥+ sup -----------
: 1 [0,1] 2 1 x≠y |x-y|
: x,y∈[0,1]
: Define M ={h∈H│∥h∥<1}, M ={f∈H│∥h∥≦1}, M ={g∈H│∥g∥≦1} and
: 0 1 1 1 2 2
: M ={u∈H│∥u∥>1}.
: 3 2
: 1)Can M_0 be open in H?
No. for any ε> 0
let f = 0 on [0,1] (i.e., f 屬於 M_0)
g(0) = n, g(ε/n) = 0, 在 (0, ε/n) 之間用直線相連, 其他為0
then d(f,g) < ε, but ∥g∥ = n > 1
0
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09/18 21:40, , 1F
09/18 21:40, 1F
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