No. Every infinite dimensional topological vector space is not locally compact. See the Wikipedia article on locally compact spaces.

In mathematics, a zero-dimensional topological space is a topological space that ... any point in the space is contained in exactly one open set of this refinement.

A topological space is simply a set, B, with topology t (see the related link for a definition), and is often denoted as B, t which is similar to how a metric space is often denoted; B, D.

There does not seem to be an under vector room, but there is vector space. Vector space is a structure that is formed by a collection of vectors. This is a term in mathematics.

Vector spaces can be formed of vector subspaces.