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Owen Lynch (Nov 17 2021 at 15:39):I feel like we need some sort of topological condition that is analogous to the completeness of Hilbert spaces in order to work with infinite-dimensional convex spaces. The reason people typically don't think about this in statistical mechanics is because they always work with convex spaces of probability measures. However, there are reasons in QM not to always work with , and so we have a general definition of Hilbert spaces. Likewise, I believe there are reasons to work with convex spaces that are not simply convex spaces of probability measures, as discussed in the other thread.
Owen Lynch (Nov 17 2021 at 15:40):I'm just going to throw out a couple ideas here for possible candidates for such a condition.
Owen Lynch (Nov 17 2021 at 15:42):One idea is to axiomatize the idea of relative entropy. I.e., consider convex spaces with a relative entropy measure such that the convex space is complete with respect to the topology induced by the relative entropy measure.