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In what condition,

$\|u_k\|_{L^\infty(0,1)}\le C$ and

$u_{k}\rightarrow0$ strongly in $L^p(0,1),\;\forall 1<p<\infty$

imply that there exists a subsequence

$u_{k_i}\rightarrow0$ strongly in $L^\infty(0,1)$?

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I cannot think of any such condition that isn't trivially equivalent to the conclusion. –  Harald Hanche-Olsen Sep 2 '12 at 6:47
    
I edited the question by adding the strongly convergent conditon in $L^p$. –  jany Sep 2 '12 at 6:59
    
Well, given that $\lVert u_k\rVert_{L^\infty}=\lim_{p\to\infty}\lVert u_k\rVert_{L^p}$, a suitable uniformity condition on the assumed $L^p$ convergence will do the trick, but once more, I'd consider that trivially equivalent to the conclusion. But by all means, I'd be quite happy to be proved wrong. –  Harald Hanche-Olsen Sep 2 '12 at 7:11
    
Yes,I agree. But I wonder that whether there are some other conditions? Thank you so much. –  jany Sep 2 '12 at 7:21
    
Equicontinuity of $u_n$ would suffice, though this does not fit so naturally with $L^p$ spaces. I am sure that you will not find any condition that implies convergence of a subsequence without also implying convergence of the entire sequence. –  user31373 Sep 8 '12 at 0:32
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