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Keywords = Bishop–Phelps

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16 pages, 337 KiB  
Article
On Bishop–Phelps and Krein–Milman Properties
by Francisco Javier García-Pacheco
Mathematics 2023, 11(21), 4473; https://doi.org/10.3390/math11214473 - 28 Oct 2023
Cited by 1 | Viewed by 1434
Abstract
A real topological vector space is said to have the Krein–Milman property if every bounded, closed, convex subset has an extreme point. In the case of every bounded, closed, convex subset is the closed convex hull of its extreme points, then we say [...] Read more.
A real topological vector space is said to have the Krein–Milman property if every bounded, closed, convex subset has an extreme point. In the case of every bounded, closed, convex subset is the closed convex hull of its extreme points, then we say that the topological vector space satisfies the strong Krein–Milman property. The strong Krein–Milman property trivially implies the Krein–Milman property. We provide a sufficient condition for these two properties to be equivalent in the class of Hausdorff locally convex real topological vector spaces. This sufficient condition is the Bishop–Phelps property, which we introduce for real topological vector spaces by means of uniform convergence linear topologies. We study the inheritance of the Bishop–Phelps property. Nontrivial examples of topological vector spaces failing the Krein–Milman property are also given, providing us with necessary conditions to assure that the Krein–Milman property is satisfied. Finally, a sufficient condition to assure the Krein–Milman property is discussed. Full article
(This article belongs to the Collection Topology and Foundations)
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