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Given a separable metric space $A$, let $P(A)$ be the space of probabilities defined on $A$ along with its Borel sigma field. One can define Prohorov metric on $P(A)$, which induces the weak topology. It seems known that $P(A)$ is part of a locally convex topological space. But I want to know what this latter space is, and how its semi-norm or even norm can be defined. Thanks.

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