Non-universal families of separable Banach spaces

TL;DR

The paper constructs specific separable Banach spaces with monotone Schauder bases that are universal for certain non-universal families, revealing nuanced structures within Banach space theory.

Contribution

It introduces a method to build Banach spaces that are universal for analytic families but not for all separable Banach spaces, extending understanding of universality.

Findings

Existence of Banach spaces with monotone Schauder bases that are universal for certain families.

Construction of spaces that are not universal for all separable Banach spaces.

Analogous results for strictly convex spaces.

Abstract

We prove that if $ C $ is a family of separable Banach spaces which is analytic with respect to the Effros-Borel structure and none member of $ C $ is isometrically universal for all separable Banach spaces, then there exists a separable Banach space with a monotone Schauder basis which is isometrically universal for $ C $ but still not for all separable Banach spaces. We also establish an analogous result for the class of strictly convex spaces.