Proper Closed Linear Subspace of Topological Vector Space is Meager

Theorem

Let $\GF \in \set {\R, \C}$.

Let $X$ be a topological vector space over $\GF$.

Let $D$ be a proper closed linear subspace of $X$.

Then $D$ is meager.

Proof

From Set is Closed iff Equals Topological Closure, we have $D^- = D$.

From Proper Linear Subspace of Topological Vector Space has Empty Interior, we then have that $\paren {D^-}^\circ = D^\circ = \O$.

Hence $D$ is nowhere dense, and in particular meager.

$\blacksquare$