Implications of an affirmative solution to the Lindenstrauss Problem

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 c ₀ is not! However, it turns out that c ₀ is a Lipschitz retract of ℓ∞ , the bidual of c ₀ . In his famous paper of 1964, Lindenstrauss asked if each 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.