# Duality of Topological Vector Spaces

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.

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.

• What do you call a "discrete topological vector space"? – Sergei Akbarov Sep 1 '19 at 15:44
• Dean, for your question 1: this is called "a space with a weak topology". These are topological vector spaces $X$ where the topology is (the weakest topology) generated by all the linear continuous functionals $f:X\to K$. This class is exactly the one where your duality holds: if $X\cong X′′$ then such a space must be a space with a weak topology by your definition of the topology on $X′′$, and on the other hand each space X with the weak topology satisfies the identity $X\cong X′′$ by the Mackey-Arens theorem (and this can be proved directly, without Mackey-Arens). – Sergei Akbarov Sep 1 '19 at 16:56
• @SergeiAkbarov I call it a "discrete topological vector space" when every subset is open. Thanks so much for your answer. – Dean Young Sep 1 '19 at 17:08
• If in $X$ every subset is open then $X$ is not a topological vector space (at least when $K={\mathbb R}$ or ${\mathbb C}$). – Sergei Akbarov Sep 1 '19 at 17:10
• Dean. this is described in the standard books on topological vector spaces, in particular, in H. Schaefer Topological vector spaces, Chapter IV (you need Proposition 1.2 in this chapter). Actually, everywhere where the author speaks about duality (meaning dual pairings) he gives this result, for example, H.Jarchow in his Locally convex spaces, Section 8.1, Theorem 2. – Sergei Akbarov Sep 2 '19 at 18:07

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.