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 (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".

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.

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.

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 (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 $M$),

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

This is proved here. 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.

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 (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".

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.