Does $A+A^\perp=\scr H$ in a Hilbert space imply $A$ is closed? This is just the converse to the Hilbert projection theorem, which says that if $A$ is a closed subspace of a Hilbert space $\scr H$ then $A+A^\perp=\scr H$. If $A$ is a linear subspace of $\scr H$ and $A+A^\perp=\scr H$, can we show $A$ is closed?
Context: Just a problem I thought up, and I don't have any work specifically tacking this problem to show. (If I did, I would post it as answer instead of cluttering the OP.)

Yes.  Suppose $x_n \in A$ and $x_n \to x$.  If $A + A^\perp = \mathcal{H}$ then we can write $x = y + z$ where $y \in A$ and $z \in A^\perp$.  That is, $x-y \in A^\perp$.  But then
$$\|x-y\|^2 = \langle x-y, x-y \rangle = \lim_{n \to \infty} \langle x_n - y, x-y \rangle = 0$$
since $x-y \in A^\perp$ and $x_n - y \in A$ for all $n$.  So $x=y$, meaning $x \in A$.
