Closed vector subspaces of large powers of R

Question

By a large power of $\mathbb R$ is meant a topological vector space which is the product of infinitely many copies of the real line.

Is every closed subspace of such a TVS linearly homeomorphic to a power of $\mathbb R$?

Answer 1

Yes, that's true. This is called a space of minimal type (see. H.H.Schaefer p.191):

A locally convex space $X$ over $\mathbb R$ is isomorphic to some ${\mathbb R}^I$ ($I$ being a cardinal number) if and only if $X$ has no weaker (Hausdorff) locally convex topology or, what is equivalent, if $X$ is weakly complete.

As a corollary, each closed subspace (and each Hausdorff quotient space) of ${\mathbb R}^I$ is isomorphic to some ${\mathbb R}^J$.