Re: [分析] 實變sequential convergene/compactness
※ 引述《BA0954016 (YES)》之銘言：
: 1. Let 1<p<∞ and f_0 belong to L^p(R). For each natural number n, define
: f_n(x)=f_0(x-n) for all x. Define f=0 on R. Show that {f_n}-->f in L^p(R). Is
: this true for p=1?
pf. Let g\in L^q(R) where 1/p+1/q=1. Then for any \epsilon > 0, there is M > 0
s.t. ||g\chi_{|x|>M}||_q \leq \epsilon.
Therefore
|< f_n, g >|\leq \epsilon ||f_n||_p + ||g||_q ||f_n\chi_{|x|>M}||_p
where ||f_n\chi_{|x|>M}||_p -> 0 as n -> \infty. Done.
When p=1, this statement fails by taking f_0 as a positive function and g=1
as a constant function.
: 2. Let [a,b] be a nondegenerate closed, bounded interval. In the Banach space
: C[a,b], normed by the maximum norm, find a bounded sequence that fails to
: have any strongly convergent subsequence.
Counter example. \chi_{(2^{-n},2^{-n+1})} (say, a=0,b=1 in our case.)
: 感謝各位實變高手能給我一些解答或方向!
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