2 added 418 characters in body; edited title; deleted 477 characters in body; added 8 characters in body

# When LCS is isomorphic to subspaceof some function space?

It is known that any vector space can be realized as some function space. Now let us put

Updated: Following Michael's suggestion, I rephrase the vector topology inquestion slightly. When will

Given a locally convex (Hausdorff) topological vector space be (LCTVS), when is it isomorphic to a subspace of some function space $Y^X$ (equipped with the product topology)? Here by function , where $Y$ is, say, some Banach space I mean (if it helps simplify things, can assume $Y=\mathbb{C}$, the vector space of complex valued functions over field), and $X$ is some set.

Apparently, only locally convex spaces (LCS) can be our candidate (for the product topology is locally convex)We are free to choose X and Y. The question is whether all LCS can be realized as some function space?

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

Any reference would be appreciated, thanks!

1

# When LCS is isomorphic to some function space?

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!