Pontryagin-reflexivity of spaces of continuous functions

Question

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.

Answer 1

This is a bit tangential to your query but I hope that it might be useful. The duality for Banach spaces that you mention is, in my opinion, best expressed in terms of the symmetric one between between Banach spaces and Waelbroeck spaces. The latter concept is due to Waelbroeck and Buchwalter and is perhaps easiest accessible in the classic text by Cigler, Losert and Michor ("Functors on Categories of Banach Spaces"). A convenient way to look at it is in the context of the pro and ind categories (Grothendick et al.). The category of Banach spaces (with linear contractions as morphisms) is the ind completion of the finite dimensional ones, that of Waelbrock spaces the pro-completion so the duality follows by abstract nonsense from the finite dimensional case.

With regard to the spaces of continuous functions, firstly I am of the opinion that the natural framework is that of (functionally separated) compactological spaces (again due to Buchwaltere, based on work by Waelbroeck) and the so-called Saks spaces. The latter are Banach spaces with a suitable auxiliary l.c. topology on the unit ball. One then considers a duality between a compactological spaces and the bounded continuous functions thereon, regarded as a Saks space with the uniform norm and the topology of compact convergence. The details are too convoluted to expose here but can be found in the monograph "Saks spaces and Applications to Functional Analysis (first edition)". Once again, this duality is natural when considered in the context of pro and ind categories (applied to the categories of compact resp. Banach spaces).

One can also presumable develop a suitable duality in terms of (not necessarily) bounded continuous functions as in your query, it is just that nobody has, to my knowledge, actually sat down and done this.

Answer 2

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.

In my opinion, the nearest area to this is the theory of stereotype spaces (in nLab it is mentioned here). Your type of reflexivity is called reflectivity and is studied in the works related to this theory, for example, here.

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, 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 $M$),

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

This is proved here. Also soon there must be published a book in De Gruyter titled "Stereotype spaces and algebras", where these questions are discussed in detail.

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 close questions here.