A general abstract approach to approximation properties in Banach spaces

TL;DR

This paper introduces a unified abstract framework for approximation properties in Banach spaces, generalizing many classical and recent results by defining the (I,J,{ au})-approximation property and ideal topologies.

Contribution

It develops a general approach that encompasses various approximation properties in Banach spaces through the (I,J,{ au})-approximation property and ideal topologies, unifying existing theories.

Findings

Recovers many classical approximation properties as special cases.

Provides a general framework that simplifies proofs of known results.

Unifies diverse approximation concepts under a single abstract approach.

Abstract

We propose a unifying approach to many approximation properties studied in the literature from the 1930s up to our days. To do so, we say that a Banach space E has the (I,J,{\tau})-approximation property if E-valued operators belonging to the operator ideal I can be approximated, with respect to the topology {\tau}, by operators belonging to the operator ideal J. Restricting {\tau} to a class of linear topologies, which we call ideal topologies, this concept recovers many classical/recent approximation properties as particular instances and several important known results are particular cases of more general results that are valid in this abstract framework.