Suppose I have a sequence of continuous function $u_n : B_1(0) \rightarrow \mathbb{R}$ and $u_n \leq \log1/r$ for all $n$. If I know that $u_n$ weakly converges to $u$ in $W^{1,2}(B_1(0))$. Could I conclude that $u_n$ converges pointwisely along some subsequence locally in $B_1(0) \setminus \{0\}$?
Weakly convergence and pointwise convergence for locally bounded function
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functional-analysis
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0$W^{1,2}$ is compactly embedded in $L^2$ and every $L^2$ convergent sequence has a subsequence which converges pointwise a.e. Does I miss something? – 2017-01-25