Closed Linear Subspaces Closed under Intersection
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Theorem
Let $V$ be a topological vector space.
Let $\family {M_i}_{i \mathop \in I}$ be an $I$-indexed family of closed linear subspaces of $V$.
Then $M := \ds \bigcap_{i \mathop \in I} M_i$ is also a closed linear subspace of $V$.
Proof
By Set of Linear Subspaces is Closed under Intersection, $M$ is a linear subspace of $V$.
By Topology Defined by Closed Sets, the intersection of closed sets is again closed.
As the $M_i$ are all closed, it follows that $M$ is closed.
Hence $M$ is a closed linear subspace of $V$.
$\blacksquare$