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If $H$ is a separable Hilbert space (say, $L^2(\mathbb{R}^d)$ for concreteness), then it is well-known that the unit sphere $S$ of $H$ is a Hilbert manifold modeled on $H$ itself. The coordinate charts are given by stereographic projection.

Suppose I consider the Borel measures on $H$, denoted by $\mathcal{M}(H)$. Let $\mathcal{P}(H)$ denote the subset of measures with measure 1 (i.e. probability measures). My question is the following.

Question. Is there a "natural" topology on $\mathcal{P}(H)$ for which there exists a "natural" smooth structure for $\mathcal{P}(H)$ as a manifold modeled on a locally convex space?

I don't have great motivation my question; it's largely just been something about which I have been curious but have been unable any treatment of which in the literature. I suppose some motivation for my question comes from symmetric states and quantum de Finetti theorems (for example, this article).

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    $\begingroup$ Some natural ways to see it as something vaguely similar to a manifold, close to what you have in mind, are described here: mathoverflow.net/questions/277567/…, using the Wasserstein metric and the Fisher-Rao metric. There are no charts but you still have "tangents" (tangent cones). $\endgroup$
    – Mizar
    Commented Jul 30, 2019 at 19:54
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    $\begingroup$ Interpreting your question literally, such a manifold structure, if it ever exists, should be quite awkward. Indeed, $\mathcal P(H)$ as a convex set has extremal points (all Dirac masses), which would all lie on the boundary of such manifold, at least if the manifold structure is "honest" enough... $\endgroup$
    – Mizar
    Commented Jul 30, 2019 at 19:58
  • $\begingroup$ I think chapter III of Čencov/Chentsov/Ченцов's book "Statistical Decision Rules and Optimal Inference" is relevant: bookstore.ams.org/mmono-53 $\endgroup$
    – Robert Furber
    Commented Jul 31, 2019 at 0:47
  • $\begingroup$ @RobertFurber Thanks for the suggesting the reference. I was not familiar with it. $\endgroup$
    – Matt Rosenzweig
    Commented Aug 2, 2019 at 16:02

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