Genericity of Fr\'echet smooth spaces

TL;DR

This paper proves that if a separable Banach space contains all separable reflexive Fréchet smooth spaces, it must contain all separable Banach spaces, extending previous results by Godefroy and Kalton.

Contribution

It establishes a new characterization of Banach spaces containing all separable Banach spaces based on embeddings of reflexive Fréchet smooth spaces.

Findings

Contains all separable Banach spaces if it contains all separable reflexive Fréchet smooth spaces.

Extends Godefroy and Kalton's result to broader classes of Banach spaces.

Provides a new perspective on the structure of Banach spaces with smoothness properties.

Abstract

If a separable Banach space contains an isometric copy of every separable reflexive Fr\'echet smooth Banach space, then it contains an isometric copy of every separable Banach space. The same conclusion holds if we consider separable Banach spaces with Fr\'echet smooth dual space. This improves a result of G. Godefroy and N. J. Kalton.