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On Preserved and Unpreserved Extreme Points

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Descriptive Topology and Functional Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 80))

Abstract

An extreme point of the closed unit ball of a Banach space is said to be preserved if it is extreme of the closed unit ball of the bidual space; otherwise it is called unpreserved. The beginning of the present work takes the form of a survey on this topic, presenting some elementary facta about those concepts—usually with new proofs—and discussing in particular Katznelson’s solution to a Phelps’ question on preserved extreme points, not available, to our knowledge, in the literature. In a second part, some new results are presented. Since some of them depend on the concept of polyhedrality, we first review several results on this topic. Then we present Godun renorming theorem for the class of nonreflexive Banach spaces, and Morris renorming result—with a new proof—on separable Banach spaces containing a copy of \(c_0\). We show that, under some extra conditions—polyhedrality—a similar renorming, this time adding smoothness, can be defined ensuring strict convexity with all points in the unit sphere unpreserved extreme. We finalize this work by presenting what—to our knowledge—is the first nonseparable result of this kind for the natural class of the weakly compactly generated Banach spaces.

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Notes

  1. 1.

    We define this, seemingly, artificial sequence \(\{A_k\}\) to allow further manipulations in subsequent arguments, although, strictly speaking, we need only here a set \(A_k\not =\{0\}\) in the sequence.

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Acknowledgments

We thank Prof. M. López Pellicer, who suggested to survey on this topic as a result of lecturing at the First Meeting in Topology and Functional Analysis, to honor Prof. J. Kąkol on the occasion of his 60th birthday. Elche, Alicante, 2013. We thank also Ministerio de Economía y Competitividad and FEDER for its support under project MTM2011-25377 (A. J. Guirao), and project MTM2011-22417 (V. Montesinos).

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Authors and Affiliations

  1. Instituto de Matemática Pura y Aplicada, Universitat Politècnica de València, C/ Vera s/n, 46022, Valencia, Spain

    Antonio José Guirao & Vicente Montesinos

  2. Department of Mathematics, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Václav Zizler

Authors
  1. Antonio José Guirao
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  2. Vicente Montesinos
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  3. Václav Zizler
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Correspondence to Vicente Montesinos .

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Editors and Affiliations

  1. Center Operations Research Univ. Inst., Universidad Miguel Hernandez, Elche, Alicante, Spain

    Juan Carlos Ferrando

  2. Matematica Aplicada, Universidad Politecnica de Valencia, Valencia, Valencia, Spain

    Manuel López-Pellicer

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Guirao, A.J., Montesinos, V., Zizler, V. (2014). On Preserved and Unpreserved Extreme Points. In: Ferrando, J., López-Pellicer, M. (eds) Descriptive Topology and Functional Analysis. Springer Proceedings in Mathematics & Statistics, vol 80. Springer, Cham. https://doi.org/10.1007/978-3-319-05224-3_9

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