Let $Q = (0,1)^d$. We define $$H^1_{per}(Q) = \{ v \in H^1(Q) | v \mbox{ is Q-periodic} \}.$$ How to prove that is a Hilbert space ? I do it easily when $d=1$ thanks to the continuity of the trace which is definite pointwise but we loose that pointwise argument when $d >1$.
I would like to show it is a closed subspace of $H^1(Q)$ using the fact that $v_n \rightarrow v$ in $L^2$ implies their exist a subsequence of $(v_n)$ converging pointwise a.e but the 'a.e' seems annoying since the $Q$-periodicity is a pointwise notion.
Thanks