On Bishop–Phelps and Krein–Milman Properties

Author:

García-Pacheco Francisco Javier1ORCID

Affiliation:

1. Department of Mathematics, College of Engineering, Avda. de la Universidad 10, 11519 Puerto Real, Spain

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 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.

Funder

Consejería de Universidad, Investigación e Innovación de la Junta de Andalucía

Publisher

MDPI AG

Subject

General Mathematics,Engineering (miscellaneous),Computer Science (miscellaneous)

Reference39 articles.

1. Banach, S. (1993). Théorie des Opérations Linéaires, Éditions Jacques Gabay. Reprint of the 1932 Original.

2. Grothendieck, A. (1955). Produits Tensoriels Topologiques et Espaces Nucléaires, American Mathematical Society. Memoirs of the American Mathematical Society.

3. Grothendieck, A. (1955). Produits Tensoriels Topologiques et Espaces Nucléaires, American Mathematical Society. Memoirs of the American Mathematical Society.

4. Grothendieck, A. (1960). Proceedings of the International Congress of Mathematicians 1958, Cambridge University Press.

5. Éléments de géométrie algébrique. I. Le langage des schémas;Grothendieck;Inst. Hautes Études Sci. Publ. Math.,1960

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