Duality of Topological Vector Spaces

Question

Let $K$ be a topological field. Let $\text{top-} K \text{-vect}$ be the category of topological $K$-vector spaces $V$, so that the maps $\cdot : K \times V \rightarrow V$ and $+ : V \times V \rightarrow V$ must be continuous in addition to the ordinary axioms.

Define a functor $[-, K]_{\text{top-}K \text{-vect}} : \text{top-}K \text{-vect}^{\text{op}} \rightarrow \text{top-}K \text{-vect}$ sending a $K$-vector space $V$ to the $K$ vector space of continuous $K$-vector space maps from $V$ to $K$. We put a topology on $[V, K]_{\text{top-}K \text{-vect}}$ which is the weakest topology such that $\hat{a} : [V, K]_{\text{top-}K \text{-vect}} \rightarrow K$, $\phi \mapsto \phi(a)$ is continuous for each $a \in V$. $[-, K]_{\text{top-}K \text{-vect}}$ is right adjoint to $[-, K]_{\text{top-}K \text{-vect}}^{op}$: $$[-, K]_{\text{top-}K \text{-vect}} : \text{top-}K \text{-vect}^{\text{op}} \leftrightarrow \text{top-}K \text{-vect} : [-, K]_{\text{top-}K \text{-vect}}^{op} $$

Question: For which topological vector spaces $V$ is the canonical map $\eta_V : V \rightarrow [[V, K]_{\text{top-}K \text{-vect}}, K]_{\text{top-}K \text{-vect}}$ an isomorphism?

I am looking for a characterization, or a broad class of examples.

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Define a functor $[-, K]_{K \text{-vect}} :K \text{-vect}^{\text{op}} \rightarrow K \text{-vect}$ sending a $K$-vector space $V$ to the $K$ vector space of $K$-vector space maps from $V$ to $K$. $[-, K]_{K \text{-vect}}$ is right adjoint to $[-, K]_{K \text{-vect}}^{op}$: $$[-, K]_{K \text{-vect}} : K \text{-vect}^{\text{op}} \leftrightarrow K \text{-vect} : [-, K]_{K \text{-vect}}^{op} $$

Let $F : K \text{-vect} \rightarrow \text{top-} K \text{-vect}$ send a $K$-vector space to the topological $K$-vector space with a certain topology (edit: to get what this topology should be, we must set $F(K)$ to be homeomorphic to $K$, and $F$ must preserve coproducts). Let $U : \text{top-} K \text{-vect} \rightarrow K \text{-vect}$ send a topological $K$-vector space to its underlying $K$-vector space. $F$ is left adjoint to $U$. Note that this makes $F^{op}$ right adjoint to $U^{op}$

We have a commutative diagram of right adjoints:

Here, I have asked whether $\eta$ is a natural isomorphism after restricting to the class of discrete topological vector spaces. Note that this is not the same as $$[-, K]_{K \text{-vect}} : K \text{-vect}^{\text{op}} \leftrightarrow K \text{-vect} : [-, K]_{K \text{-vect}}^{op}$$ being a categorical equivalence.

Answer 1

There seems to be some confusion about what you want but here is a try. Suppose that we have two vector spaces $E$ and $F$ with a bilinear form $\phi$ on their product which separates them (obvious definition). Then we can supply $E$ with the weak topology $\sigma (E,F)$ and $F$ correspondingly. The dual of $E$ is naturally identifiable with $F$ and vice versa. Hence $E$ with this weak topology satisfies your condition. On the other hand, any such space arises in this way.

Remarks:

1) This is for the case of the real or complex field.

2) A detailed account can be found in Bourbaki („Espaces vectoriels topologiques“).