Let $ \Omega \subset \mathbb{R}^n$ be open and bounded. For $u, v \in C^1(\overline{\Omega}) $, define $$ (u,v)_{H^1} = \int_{\Omega} u(x)v(x)dx + \int_{\Omega}\nabla u(x)\cdot \nabla v(x)dx. $$ I want to show that $(C^1(\Omega), (\cdot,\cdot)_{H^1})$ is an inner product space but not complete. I've already proved that $(u,v)$ is an inner product function, but I have no clue on how to prove that it is not complete.
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1Quick beginner guide for asking a well-received question + please avoid "no clue" questions: show your attempts – Anne Bauval Dec 17 '23 at 14:49
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What have you tried? – Lorago Dec 17 '23 at 15:00
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Try for $n=1$ and $\Omega=(0,1)$ – Anne Bauval Dec 17 '23 at 16:02
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Take an easy nondifferentiable function like $|x|$ and approximate with smooth functions. – whpowell96 Dec 17 '23 at 16:15
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https://math.stackexchange.com/questions/114070/how-to-prove-that-ck-omega-is-not-complete?rq=1 – K.defaoite Dec 17 '23 at 18:47
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It is somehow alike to the sobolev space with a different norm. And the form of its norm is alike to the norm of $L^p$, my idea is trying to construct a continuous function sequence converges in norm while its limit function is not continuous. – Slime the great Dec 18 '23 at 01:08