After looking a bit at uniform spaces, as their general definition seems relevant to the study of topological vector spaces, it seems that they provide just enough structure to define the notion of total boundedness:
Say a set $B$ is totally bounded if for any entourage $V$, there are finitely many points $x_n$ such that $B \subseteq \bigcup_n V[x_n]$.
I haven't seen this definition explicitly, but I have seen a special case of it applied in the context of TVS, though not named as such. However, I can't seem to get a similar definition for boundedness proper. Specifying that $B$ be contained in some $V[x]$ just won't work, as the $V$ could be made arbitrarily large.
Is it possible to get such a definition of boundedness in uniform spaces?