  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.
