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

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 $5$).