10
$\begingroup$

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 $\mathbb R$?

$\endgroup$
8
$\begingroup$

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 ${\mathbb R}^J$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.