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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!

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  • $\begingroup$ 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. $\endgroup$ – Qiaochu Yuan Jan 4 '13 at 21:38
  • $\begingroup$ @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. $\endgroup$ – Tim Jan 4 '13 at 21:39

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