Closed Linear Subspaces Closed under Intersection

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Theorem

Let $V$ be a topological vector space.

Let $\left({M_i}\right)_{i \in I}$ be an $I$-indexed collection of closed linear subspaces of $V$.


Then $M := \displaystyle \bigcap_{i \mathop \in I} M_i$ is also a closed linear subspace of $V$.


Proof

By Linear Subspaces 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 $V$.

$\blacksquare$