I would say that this is not well-known:

> 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).

Usually, when I say this, people are surprised, ask again, and don't really trust. 

I would say the nearest area to this is the theory of [stereotype spaces][1] (in nLab it is mentioned [here][2]). Your type of reflexivity is called *reflectivity* and is studied in the works related to this theory, for example, [here][3]. 

It is known that all Fréchet spaces are stereotype and, as a corollary, for $\sigma$-compact topological spaces $M$ the spaces ${\mathcal C}(M)$ of continuous functions (with the compact-open topology) are stereotype and satisfy your reflexivity condition (i.e. *reflectivity*). Moreover, as it was explained [here][4], this is true for all paracompact locally compact spaces $M$. 

Apart from the spaces ${\mathcal C}(M)$, there are many other functional spaces that are stereotype, in fact all the functional spaces in geometry:

-  ${\mathcal E}(M)$ (the space of smooth functions on a smooth manifold $M$),

-  ${\mathcal O}(M)$ (the space of holomorphic functions on a Stein manifold $M$),

-  ${\mathcal P}(X)$ (the space of polynomials on an affine algebraic manifold $X$).

This is proved [here][4]. Also soon there must ba published a book in De Gruyter named "Stereotype spaces and algebras".

For arbitrary $k$-spaces $M$, as far as I remember, the spaces ${\mathcal C}(M)$ are not necessarily reflective (and not necessarily stereotype), but I can't recall a reference now. I remember that Salvador Hernandez and Vladimir Uspenskij studied closed questions [here][5].


  [1]: https://handwiki.org/wiki/Stereotype_space
  [2]: https://ncatlab.org/nlab/show/stereotype+space
  [3]: https://zbmath.org/?q=an:1076.22001
  [4]: https://zbmath.org/?q=an:1042.46002
  [5]: https://in.art1lib.com/book/1934327/045ba5