# 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.

$\blacksquare$