[分析] function space

[jacky7987 (憶)]

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?
2)Can M_1 be closed in H?
3)Can M_2 be closed in H?
4)Can M_3 be open in H?

我主要不行的是
1.找不到適合的半徑讓他落在集合裡面
2.無法在這些距離和norm中間自由的轉換

不確定是不是四題的做法都類似
如果差不多的話可以作其中一題 我剩下會自己想辦法
感謝QAQ

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※ 編輯: jacky7987       來自: 111.240.240.66       (09/18 17:09)