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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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    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

0 Answers 0