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Let $2 \le p<\frac{2n}{n-2}$. Suppose that a sequence $\{u_k\}_k\subset H^1(\mathbb{R}^n)$ weakly converges to $u \in H^1(\mathbb{R}^n)$, and hence weakly converges to $u$ in $L^p(\mathbb{R}^n)$. How can I prove that $$ ||u_k||_{H^i}-||u_k-u||_{H^i} \rightarrow ||u||_{H^i}, \ \ \ \ \ ||u_k||_{L^p}-||u_k-u||_{L^p} \rightarrow ||u||_{L^p} $$ as $k$ increases?

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What is $H^i{}$? –  Davide Giraudo Apr 1 '13 at 11:38
    
$H^1$ stands for the Sobolev space $W^{1,2}(\mathbb{R}^n)$. –  Corry Apr 1 '13 at 12:48
    
Ok, but why the letter $i$? –  Davide Giraudo Apr 1 '13 at 12:50
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