Transfinite (almost isometric) ideals in Banach spaces

TL;DR

This paper introduces and analyzes transfinite versions of (almost isometric) ideals in Banach spaces, linking them to classical spaces like Lindenstrauss and Gurari spaces, and provides new characterizations and examples.

Contribution

It develops a framework for transfinite ideals in Banach spaces, extending classical notions and offering new characterizations and examples beyond traditional spaces.

Findings

Characterization of transfinite injective spaces

Construction of examples outside classical Banach spaces

Revision of classical results for transfinite ideals

Abstract

We present and study some transfinite versions of (almost isometric) ideals in Banach spaces. As these notions are closely related with Lindenstrauss and Gurari\u{\i} spaces respectively, we will present a similar characterization for transfinite injective spaces and spaces of (almost) universal disposition in terms of these transfinite ideals. Furthermore, we construct several examples outside these type of Banach spaces and make a revision of some classical results for transfinite ideals.