Let $X$ be a inner product space, with orthogonal decomposition $X=V \oplus W$. Give $X$ the topology induced by the norm induced by the inner product. Let $E\subset X$ be a subspace such that the projections to $V, W$ (call them $\pi_V(E), \pi_W(E)$) are closed. Does this imply that $E$ is closed itself?
Note that $X$ may not be complete; in fact I am mainly interested in the case where it is not complete. Any kind of help would be appreciated!