On page 20 of these lectures, Peter Scholze and Dustin Clausen show that a broad class of topological vector spaces is reflexive, i.e. $V \cong [[V, \mathbb{R}], \mathbb{R}]$, where we endow these hom-spaces with the compact-open topology.
I was thinking that surely it was known that these classes of topological vector spaces were reflexive long before condensed mathematics. My question is, what are broad classes of topological vector spaces which are reflexive with respect to $\mathbb{R}$ under the compact open topology, and references to proofs of this fact?
I am especially interested in compactly generated spaces.