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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$\begingroup$ I cannot think of any such condition that isn't trivially equivalent to the conclusion. $\endgroup$– Harald Hanche-OlsenSep 2, 2012 at 6:47
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$\begingroup$ I edited the question by adding the strongly convergent conditon in $L^p$. $\endgroup$– janySep 2, 2012 at 6:59
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$\begingroup$ 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. $\endgroup$– Harald Hanche-OlsenSep 2, 2012 at 7:11
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$\begingroup$ Yes,I agree. But I wonder that whether there are some other conditions? Thank you so much. $\endgroup$– janySep 2, 2012 at 7:21
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$\begingroup$ 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. $\endgroup$– user31373Sep 8, 2012 at 0:32
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