On the definition of "almost LUR (ALUR)" notion

TL;DR

This paper critically examines the definition and proofs related to the almost LUR (ALUR) notion in Banach spaces, identifies gaps, and proposes a refined limit approach to clarify the concept.

Contribution

It analyzes existing works on ALUR, highlights proof gaps, and introduces a new limit definition using liminf to improve rigor.

Findings

Identified gaps in proofs involving ALUR definitions.

Proposed replacing double limits with liminf for clarity.

Provided complete proofs using the new limit approach.

Abstract

The notion of almost LUR (ALUR) point is introduced in the paper [P. Bandyopadhyay et al., Some generalizations of locally uniform rotundity, J. Math. Anal. Appl., 252, 906-916 (2000)], where one says that the point $x$ of the unit sphere $S_{X}$ of a Banach space is an almost LUR (ALUR) point of $B_{X}$ if for any sequences $\{x_{n}\}\subseteq B_{X}$ and $\{x_{m}^{\ast}\}\subseteq B_{X^{\ast}}$, the condition $\lim_{m}\lim_{n}x_{m}^{\ast}\left(\frac{x_{n}+x}{2}\right)=1$ implies $\lim_{n}\left\Vert x_{n}-x\right\Vert =0$, without mentioning what is meant by $\lim_{m}\lim_{n}\gamma_{m,n}=\gamma$ for $\gamma$, $\gamma_{m,n}\in\mathbb{R}$; $X$ is ALUR if $X$ is almost LUR at any $x\in S_{X}$. Of course, the natural definition for this iterated limit would be that for each $m$ sufficiently large there exists $\mu_{m}:=\lim_{n\rightarrow\infty}\gamma_{m,n}\in\mathbb{R}$ and $\gamma =\lim_{m\rightarrow\infty}\mu_{m}$. However, as seen in some proofs where $\lim_{m}\lim_{n}$ appears, this interpretation is not confirmed. In this paper we examine several works in which almost LUR is mentioned and, especially, the proofs of those results in which the above definition of "almost LUR" point (or space) is invoked. Moreover, we analyze similar problems related to the notion CWALUR which extend ALUR. Furthermore, we mention several gaps in the proofs of some results. Finally, we propose the change of $\lim_{m}\lim_{n}$ by $\lim_{m}\liminf_{n}$ in the definitions of several types of ALUR points; moreover, we provide the complete proofs of two results from the literature in which $\lim_{m}\lim_{n}$ were used effectively, using $\lim_{m}\liminf_{n}$ instead.