Orthogonal Difference is Closed Linear Subspace
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Theorem
Let $H$ be a Hilbert space.
Let $M, N$ be closed linear subspaces of $H$.
Then the orthogonal difference $M \ominus N$ is also a closed linear subspace of $H$.
Proof
By definition, $M \ominus N = M \cap N^\perp$.
By Orthocomplement is Closed Linear Subspace, $N^\perp$ is a closed linear subspace of $H$.
Hence the result, by Closed Linear Subspaces Closed under Intersection.
$\blacksquare$