L-Orthogonal Elements and L-Orthogonal Sequences

Author:

Avilés Antonio1,Martínez-Cervantes Gonzalo2,Rueda Zoca Abraham1

Affiliation:

1. Departamento de Matemáticas, Campus de Espinardo, Universidad de Murcia, 30100 Murcia, Spain

2. Departamento de Matemáticas, Facultad de Ciencias, Universidad de Alicante, 03080 Alicante, Spain

Abstract

Abstract Given a Banach space $X$, we say that a sequence $\{x_n\}$ in the unit ball of $X$ is $L$-orthogonal if $\Vert x+x_n\Vert \rightarrow 1+\Vert x\Vert $ for every $x\in X$. On the other hand, an element $x^{**}$ in the bidual sphere is said to be $L$-orthogonal (to $X$) if $\|x+x^{**}\|= 1+\Vert x\Vert $ for every $x\in X$. The aim of this paper is to clarify the relation between $L$-orthogonal sequences and $L$-orthogonal elements. Namely, we study whether every $L$-orthogonal sequence contains $L$-orthogonal elements in its weak*-closure. We provide an affirmative answer whenever the ambient space has small density character. Nevertheless, we show that, surprisingly, the general answer is independent of the usual axioms of set theory.

Funder

Government of Spain, AEI/FEDER, EU

Fundación Séneca, ACyT Región de Murcia

European Social Fund

Youth European Initiative

Juan de la Cierva-Formación fellowship

Spanish AEI Project

Junta de Andalucía

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

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