The Dual Form of the Approximation Property for a Banach Space and a Subspace

TL;DR

This paper introduces a dual formulation of the approximation property for Banach spaces and subspaces, enabling easier analysis and applications, such as inheritance of the property by quotient spaces.

Contribution

It provides a new dual characterization of the approximation property for Banach spaces with subspaces, facilitating applications to related space properties.

Findings

Dual formulation simplifies checking the approximation property.

If X has AP and Y is L-infinity, then X/Y also has AP.

Application to three space properties enhances understanding of Banach space structure.

Abstract

Given a Banach space X and a subspace Y, the pair (X,Y) is said to have the approximation property (AP) provided there is a net of finite rank bounded linear operators on X all of which leave the subspace Y invariant such that the net converges uniformly on compact subsets of X to the identity operator. The main result is an easy to apply dual formulation of this property. Applications are given to three space properties; in particular, if X has the approximation property and its subspace Y is script L-infinity, then X/Y has the approximation property.