The Existence of Linear Selection and the Quotient Lifting Property

TL;DR

This paper investigates lifting properties in Banach spaces, introduces an alternative lifting property, and characterizes the quotient lifting property for pairs of Banach spaces with proximinal subspaces.

Contribution

It proposes an alternative version of the lifting property and provides a characterization of the quotient lifting property for specific Banach space pairs.

Findings

An alternative lifting property is introduced.

Conditions for the quotient lifting property are established.

A characterization of the property up to isomorphism is provided.

Abstract

Lifting properties for Banach spaces are studied. An alternate version of the lifting property due to Lindenstrass and Tzafriri is proposed and a characterization, up to isomorphism, is given. The quotient lifting property for pairs of Banach spaces $(X,J)$, with $J$ proximinal in $X$, is considered and several conditions for the property to hold are given.