Let $\Omega$ be an open subset of $\mathbb R^k$. It is well-known that $L^2(\Omega)$ is a Hilbert space. But is the following local $L^2$-integrable function space $$ L^2_\text{loc}(\Omega):=\{ f:\Omega\to \mathbb C \ \text{measurable} \mid f\in L^2(K) \ \ \ \forall K \subset \subset \Omega \} $$ still a Hilbert space? How about $W^{k,2}_\text{loc}(\Omega)$(defined in a similar way)?
More generally, what can we say about $W^{k,p}_\text{loc}(\Omega)$?