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In their book Topological Vector Spaces (2nd ed.) Lawrence Narici and Edward Beckenstein generalise convex sets for TVS over ultravalued field $K$ as $K$-convex sets. The definition goes as following:...

This is a cross-post from MSE: Elementary applications of Krein-Milman. I'm starting to suspect that the question just doesn't really have a great answer, it's worth a try. Recall that the Krein-...