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!