I know that all real, finite-dimensional topological vector spaces are isomorphic to $\mathbb{R}^n$ for some $n$, but are they also homeomorphic?

The reason I'm asking this is because I was wondering whether or not there were any disconnected real topological vector spaces.

nottrivial. It is easier if you require in addition local convexity. André Weil proves the general fact in one of the very first sections of his "Basic Number Theory". $\endgroup$ – Theo Buehler Dec 25 '10 at 8:45thatdifficult (essentially it boils down to compactness of the unit ball in standard $\mathbb{R}^{n}$). @Harry: No, uniform structures do not enter explicitly (but they lurk around, of course). Yes, uniform structures were invented by Weil, they arose in his investigations of topological groups in the late 30's. $\endgroup$ – Theo Buehler Dec 25 '10 at 11:53