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Norming Markushevich bases: recent results and open problems

Petr Hájek, Tommaso Russo

Abstract

We survey several results concerning norming Markushevich bases (M-bases, for short), focusing in particular on two recent examples of a weakly compactly generated Banach space with no norming M-basis and of an Asplund space with norming M-basis that is not weakly compactly generated. We highlight the context for these problems and state several open problems in different directions that arise from these results.

Norming Markushevich bases: recent results and open problems

Abstract

We survey several results concerning norming Markushevich bases (M-bases, for short), focusing in particular on two recent examples of a weakly compactly generated Banach space with no norming M-basis and of an Asplund space with norming M-basis that is not weakly compactly generated. We highlight the context for these problems and state several open problems in different directions that arise from these results.
Paper Structure (7 sections, 14 theorems, 50 equations, 1 figure)

This paper contains 7 sections, 14 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Examples of M-bases
  3. Classes of non-separable Banach spaces and M-bases
  4. Norming M-bases and WCG spaces
  5. Banach spaces of continuous functions
  6. The heredity problem
  7. Semi-Eberlein compacta

Key Result

Lemma 2.4

Let $\{e_k;\varphi_k\}_{k=1}^\infty$ be a norming M-basis for $\mathcal{X}$. Then some subsequence of $(e_k)_{k=1}^\infty$ is a basic sequence in $\mathcal{X}$.

Figures (1)

  • Figure 1: The function $z_N$ for $N=4$.

Theorems & Definitions (46)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5: Markushevich
  • Remark 2.6
  • Remark 4.2: JZ Some notes WCG
  • Theorem 4.3
  • Theorem 4.5: Hajek
  • ...and 36 more