@MattN you want to assume a few more conditions at least: local convexity is certainly necessary and you should maybe clarify what kinds of characterizations would be acceptable... Most of them will be rather tautological, I'd say, for example: the map $X \to \mathbb{R}^{X'}$ is a topological embedding. Or it's the weakest topology compatible with the dual pairing $(X,X')$, or: it is the $\mathfrak{S}$-topology induced by the bounded subsets contained in finite-dimensional subspaces of $X'$.
