It is known that any vector space can be realized as some function space. Now let us put the vector topology in. When will a topological vector space be isomorphic to some function space (equipped with the product topology)? Here by function space I mean the vector space of complex valued functions over some set.

Apparently, only locally convex spaces (LCS) can be our candidate (for the product topology is locally convex). The question is whether all LCS can be realized as some function space? If not, what kind of conditions should we put?

Any reference would be appreciated, thanks!

Updated: Following Michael's suggestion, I rephrase the question slightly.

Given a locally convex (Hausdorff) topological vector space (LCTVS), when is it isomorphic to a subspace of some function space $Y^X$ (equipped with the product topology), where $Y$ is, say, some Banach space (if it helps simplify things, can assume $Y=\mathbb{C}$, the complex field), and $X$ is some set. We are free to choose X and Y.

If not all LCTVS have this property, then what kind of conditions do we need?

Any reference would be appreciated, thanks!