Genericity of Fréchet smooth spaces

Abstract

If a separable Banach space X contains an isometric copy of every separable reflexive Fréchet smooth Banach space, then X contains an isometric copy of every separable Banach space. The same conclusion holds if we consider separable Banach spaces with Fréchet smooth dual space.

This improves a result of G. Godefroy and N.J.

Kalton.

Keywords

• Frechet smoothness
• Isometrically universal Banach space
• Monotone basis
• Effros-Borel structure
• Well-founded tree