Duality on Banach spaces and a Borel parametrized version of Zippin's theorem

TL;DR

This paper establishes a Borel measurable duality operation for separable Banach spaces with separable duals and provides a Borel parametrized version of Zippin's theorem, enhancing the descriptive set-theoretic understanding of Banach space theory.

Contribution

It introduces a Borel measurable duality map and a Borel parametrized embedding theorem for separable Banach spaces, extending classical results with descriptive set theory techniques.

Findings

Borel assignment of dual spaces for certain Banach spaces.

Existence of a universal Banach space with Borel parametrized embeddings.

Enhanced understanding of the structure of Banach spaces via Borel functions.

Abstract

Let SB be the standard coding for separable Banach spaces as subspaces of $C(\Delta)$. In these notes, we show that if $\mathbb{B} \subset \text{SB}$ is a Borel subset of spaces with separable dual, then the assignment $X \mapsto X^*$ can be realized by a Borel function $\mathbb{B}\to \text{SB}$. Moreover, this assignment can be done in such a way that the functional evaluation is still well defined (Theorem $1$). Also, we prove a Borel parametrized version of Zippin's theorem, i.e., we prove that there exists $Z \in \text{SB}$ and a Borel function that assigns for each $X \in \mathbb{B}$ an isomorphic copy of $X$ inside of $Z$ (Theorem $5$).