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!



When LCS is isomorphic to subspace of some function space?
If I interpret the question correctly, Yaoliang would like to know which LCTVS are isomorphic to $\mathbb{C}^X$, where $X$ is a set (no topology), and $\mathbb{C}^X$ is given the product topology. If this is so, the answer is: very few. Actually, spaces of this form are fully determined by the cardinality of $X$.


Just to make an example: no infinite dimensional normed space can be isomorphic to a space of this form. Indeed, in this case the set $X$ would have to be infinite, but then every neighborhood of $0$ in $\mathbb{C}^X$ would contain a proper vector subspace, and this is never true for normed spaces.


