It is well-known that a Banach space $V$ is always Pontryagin-reflexive, i.e. the natural map $V\to \text{Hom}_\mathbb{R}(\text{Hom}_\mathbb{R}(V, \mathbb{R}), \mathbb{R})$, where $\text{Hom}_\mathbb{R}(-, \mathbb{R})$ is endowed with the compact-open topology, is an isomorphism (of topological vector spaces). In particular, if $X$ is a compact Hausdorff space, this applies to the Banach space $V=C(X)=\mathbb{R}^X$ which is also the exponential in the category of $k$-spaces (compactly generated (weak) Hausdorff spaces). So a natural question is:
What about non-compact $k$-spaces $X$? Is the space of continuous functions $C(X)$ (endowed with the exponential, i.e. compact-open topology) Pontryagin-reflexive?
Generalizing away from $k$-spaces, can we at least find some convenient category of spaces (in the technical sense) $S$ such that the natural map $$ \mathbb{R}^X \to \text{Hom}_\mathbb{R}(\text{Hom}_\mathbb{R}(\mathbb{R}^X, \mathbb{R}), \mathbb{R}) \;\;\;\;\;(X\in S)$$ is always an isomorphism? (Where $\mathbb{R}^X$ is the exponential in $S$ and the $\text{Hom}$-sets carry the subspace topology induced from the respective exponential.)
This seems like a very natural question to me. Yet, a brief literature search yielded some results concerning characterizations of Pontryagin-reflexivity of topological vector spaces, but nothing that I was able to directly apply to this question. Is there anything known about this?
EDIT: By $\text{Hom}_\mathbb{R}(-, -)$ I mean continuous linear maps, indeed.