When LCS is isomorphic to subspace of some function space? - MathOverflow

When LCS is isomorphic to subspace of some function space?

2012-04-29T00:43:51Z

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!

Answer by Nik Weaver for When LCS is isomorphic to subspace of some function space?

2012-04-29T02:37:08Z

Yes, every LCTVS can be realized as a function space: every LCTVS is isomorphic to the dual of its dual space equipped with the weak* topology. This is a consequence of a version of the Hahn-Banach theorem which states that in a real LCTVS two disjoint closed convex sets, one of which is compact, can be strictly separated. See Section IV.3 of A Course in Functional Analysis by Conway.

Answer by Alberto Abbondandolo for When LCS is isomorphic to subspace of some function space?

2012-04-29T12:39:38Z

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.