Just wondering if $H^{1}[0,1]$ is a closed subset of $L^{2}[0,1]$. Thanks.
2026-04-28 08:34:24.1777365264
Is Sobolev space $H^{1}[0,1]$ closed w.r.t. $L^{2}$ norm?
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$C_{0}^{\infty}(0,1)\subset H^1(0,1) \subset L^2(0,1) \Rightarrow L^2(0,1)=\overline{C_{0}^{\infty}(0,1)}^{\|\cdot\|_{L^2}}\subset \overline{H^1(0,1)}^{\|\cdot\|_{L^2}} \subset L^2(0,1) \Rightarrow \overline{H^1(0,1)}^{\|\cdot\|_{L^2}} = L^2(0,1)\neq H^1(0,1)$