What is the uniform structure on a topological vector space?

Question

I've read that topological vector spaces are canonically uniform spaces, however the definitions of uniform space are quite cryptic to me! Is there a simple/intuitive way to understand why we have good notions of Cauchy sequence/uniform convergence/uniform continuity on topological vector spaces?

Answer 1

The standard way (AFAIK) to do this, is to take a local base of neighbourhoods $\mathscr{B}$ for $0$ in the TVS $X$. Then define the uniformity $\mathscr{U}$ generated by all entourages $B^\ast := \{(x,y) \in X^2: y-x \in B\}$ for every element $B$ from the local base, and showing it is family is a base for a uniformity compatible with the topology on $X$. This should be proved in any decent text on TVS's I think.