Two different opinions on whether a topological vector space is a uniform space

Van de Vel's Theory on Convexity Structures says a TVS is uniform iff it is locally convex:

3.10.1. Proposition. Let $X$ be a topological vector space, equipped with the standard convexity and with the canonical translation-invariant uniformity. Then $X$ is uniform iff it is locally convex. If, in addition, $X$ is (topologically) metrizable, then it is metrizable as a convex structure.

Wikipedia says a TVS is uniform:

a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure (a uniform structure to be precise) with the algebraic concept of a vector space.

So I guess I mess things up again. What am I missing?

Thanks and regards!

• I do not know what the first reference means by "is uniform" but it clearly isn't "is equipped with a uniformity" because that is already stated to be the case in the hypotheses. The second reference is correct. More generally, every topological group has a canonical uniformity on it. – Qiaochu Yuan Jan 4 '13 at 21:38
• @QiaochuYuan: You seem right. Two possibilities: (1) Is " the canonical translation-invariant uniformity" the one induced by the TVS itself? (2) Can "is uniform" mean "is convexity-compatibly uniform"? I don't quite understand the proof, but it seems possible. – Tim Jan 4 '13 at 21:39