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Rotund Gateaux smooth norms which are not locally uniformly rotund

Carlo Alberto De Bernardi, Jacopo Somaglia

Abstract

We provide, in every infinite-dimensional separable Banach space, an average locally uniformly rotund (and hence rotund) Gateaux smooth renorming which is not locally uniformly rotund. This solves an open problem posed by A.J. Guirao, V. Montesinos, and V. Zizler.

Rotund Gateaux smooth norms which are not locally uniformly rotund

Abstract

We provide, in every infinite-dimensional separable Banach space, an average locally uniformly rotund (and hence rotund) Gateaux smooth renorming which is not locally uniformly rotund. This solves an open problem posed by A.J. Guirao, V. Montesinos, and V. Zizler.
Paper Structure (4 sections, 15 theorems, 36 equations)

This paper contains 4 sections, 15 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Main results
  4. Consequences in nonseparable Banach spaces

Key Result

Theorem 1

Every infinite-dimensional separable Banach space admits an average locally uniformly rotund (and hence rotund) Gâteaux smooth equivalent norm $|\cdot|$ which is not locally uniformly rotund.

Theorems & Definitions (29)

  • Theorem 1
  • Lemma 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Example 2.5
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • ...and 19 more