Implications of an affirmative solution to the Lindenstrauss Problem

TL;DR

This paper explores the implications of positively resolving the Lindenstrauss Problem, linking it to key open questions in Lipschitz theory of Banach spaces and suggesting potential solutions.

Contribution

It demonstrates how an affirmative solution to the Lindenstrauss Problem could resolve several longstanding open questions in Lipschitz Banach space theory.

Findings

Lindenstrauss Problem's positive solution relates to other open Lipschitz questions.

An affirmative answer could settle multiple open problems in the field.

The paper establishes connections between the Lindenstrauss Problem and broader Lipschitz theory questions.

Abstract

The question regarding the location of Banach spaces inside their biduals has been investigated and answered reasonably satisfactorily in the linear theory of Banach spaces. Thus, for instance, whereas it is known that a dual Banach space is complemented inside its bidual, the space of all null sequences is not! However, the latter space is a Lipschitz retract of its bidual. In his famous paper of 1964, Lindenstrauss asked if every Banach space is a Lipschitz retract of its bidual. In this short note, we show how to relate the Lindenstrauss problem (LP) to certain other important and well-known questions that remain open in the Lipschitz theory of Banach spaces and how these latter questions may be settled in the affirmative under the assumption of (LP) having a positive solution.