Consider the weighted $L^2-$space defined by
$M:=\{z:\mathbb{R}^+\to H^2(\Omega)\cap H^1_0(\Omega);\int_{0}^{\infty}g(s)\|\nabla z(s)\|^2_2ds<\infty\},$
equipped with the inner product
$\langle z_1,z_2\rangle=\int_{\Omega}\int_{o}^{\infty}g(s)\nabla z_1(s).\nabla z_2(s)dsdx.$
where $g$ is a positive, non-increasing and $L^1(0,\infty)$ function.
I want to show that $M$ is complete.
This is my attempt, Let $(u_n)$ be a Cauchy sequence in $M$, this implies that $(u_n)$ is Cauchy in $H^2(\Omega)\cap H^1_0(\Omega)$. But $H^2(\Omega)\cap H^1_0(\Omega)$ is complete, therefore, $u_n\to u\in H^2(\Omega)\cap H^1_0(\Omega).$ But I could not satisfy the condition $\int_{0}^{\infty}g(s)\|\nabla u(s)\|_2^2ds<\infty$.