Topological structure of non-separable sigma-locally compact convex sets

Abstract

For an infinite cardinal $\kappa$ let $\ell_2(\kappa)$ be the linear hull of the standard othonormal base of the Hilbert space $\ell_2(\kappa)$ of density $\kappa$. We prove that a non-separable convex subset $X$ of density $\kappa$ in a locally convex linear metric space if homeomorphic to the space (i) $\ell_2^f(\kappa)$ if and only if $X$ can be written as countable union of finite-dimensional locally compact subspaces, (ii) $[0,1]^\omega\times \ell_2^f(\kappa)$ if and only if $X$ contains a topological copy of the Hilbert cube and $X$ can be written as a countable union of locally compact subspaces.