Let $\{e_k | e_k(x)= e^{ikx}/\sqrt{2\pi\,}\}$ be the orthonormal basis in $\mathcal L^2$ per. I first have to use this basis define two infinite dimensional orthogonal subspaces of $\mathcal L^2$ per.
Then I have to show whether the completeness relation holds for: $A = \{e_k | k \in\Bbb Z \setminus 3\Bbb Z\}$?
What would the two infinite dimensional orthogonal subspaces be in $\mathcal L^2$ per?
How do I prove that the completeness relation holds for $A$?
For 1, write $\mathbb{Z}$ as the union of two disjoint infinite subsets, $\mathbb{Z} = A \cup B$ and look at the subspaces spanned by $\{ e_k \}_{k\in A}$ and $\{ e_k \}_{k\in B}$.
For 2, think some more about 1.