Closed Linear Subspaces Closed under Setwise Addition

Theorem
Let $H$ be a Hilbert space.

Let $M, N$ be closed linear subspaces of $H$.

Then $M + N$ is also a closed linear subspace of $H$, where $+$ denotes setwise addition.

Proof
By Linear Subspaces Closed under Setwise Addition, $M + N$ is a linear subspace of $H$.

Now to show that it is closed.

Let $P: H \to H$ denote the orthogonal projection on $M$.

Denote by $I - P$ the complementary projection of $P$.

Define $N' := \set {n - P n: n \in N}$.

$N'$ is a closed linear subspace of $H$.

Observe $m + n = \paren {m + P n} + \paren {n - P n} \in M + N'$; hence, $M + N \subseteq M + N'$.

By $m + \paren {n - P n} = \paren {m - P n} + n \in M + N$, conclude that $M + N' \subseteq M + N$, hence equality.

Furthermore, $N' \subseteq \map {\operatorname{ran} } {I - P} = \map \ker P$ by Range of Idempotent is Kernel of Complementary Idempotent.

That is, $N' \subseteq M^\perp$ by Properties of Orthogonal Projection, and hence $M \perp N'$.

Denote by $P'$ the orthogonal projection on $N'$.

Suppose now that $h \in M + N'$. Then:

For this last equality, observe $M \perp N'$, hence $M \subseteq N'^\perp$, $N' \subseteq M^\perp$ and $\paren {M + N'}^\perp \subseteq M^\perp \cup N'^\perp$.

The conclusion is that every $h \in M + N'$ can be uniquely decomposed as $P' h + P h$, with $P' h \in N', P h \in M$.

So suppose there is a sequence $h_n \to h$ in $M + N'$. Then $P h_n$ and $P' h_n$ are sequences in $M, N'$, respectively.

As those are closed linear subspaces of $H$, there are $m \in M, n \in N'$ with $P h_n \to m, P' h_n \to n$.

It follows that $h = m + n \in M + N'$.

That is, $M + N' = M + N$ is a closed linear subspace of $H$.