Are there any known characterizations of compactly generated spaces among all topological vector spaces? Certainly all finite-dimensional vector spaces are compactly generated (because they are locally compact). More generally, all first-countable topological vector spaces are compactly generated. I would like to know if there are more, and in particular how to check efficiently if given topological vector space is compactly generated. Results valid only for special clases of TVSes (e.g. locally convex) are certainly welcome.
For reference, topological space $X$ is said to be compactly generated if subset $C \subseteq X$ is closed if and only if $C \cap K$ is closed for every compact subspace $K \subseteq X$. Secondly, I don't include Hausdorffness in the definition of a compact space. However results valid only under this additional assumption are no less interesting.