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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

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.2012-09-02
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    I edited the question by adding the strongly convergent conditon in $L^p$.2012-09-02
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    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.2012-09-02
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    Yes,I agree. But I wonder that whether there are some other conditions? Thank you so much.2012-09-02
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    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*.2012-09-08

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