Definability under duality

TL;DR

This paper proves that the class of dual Banach spaces derived from an analytic class of separable Banach spaces with separable duals is itself analytic, highlighting a specific property of duality in Banach space theory.

Contribution

It establishes the analyticity of the dual class for a given analytic class of Banach spaces with separable duals, but shows the analogous result does not hold for pre-duals.

Findings

Dual classes preserve analyticity under certain conditions

Pre-dual classes do not necessarily preserve analyticity

Provides insight into the structure of Banach space duality

Abstract

It is shown that if $A$ is an analytic class of separable Banach spaces with separable dual, then the set $A^*=\{Y:\exists X\in A \text{with} Y\cong X^*\}$ is analytic. The corresponding result for pre-duals is false.