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 $L^\infty(0,1)$?
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 $L^\infty(0,1)$?