 Closed vector subspace

A closed vector subspace of a topological vector space is both a vector subspace and a closed set. If a continuous linear map has a Hausdorff target, its kernel of a linear map is a closed vector subspace, since it is the preimage of the closed singleton $\{0\}$. The target condition matters because our topological vector space convention permits non-Hausdorff spaces.

For a counterexample without the target hypothesis, the identity linear map of $\mathbb{R}$ with the indiscrete topology is continuous, but its kernel of a linear map $\{0\}$ is not closed.