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

Braga, Bruno de Mendonça

Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2413-2435.

Soit $\text{SB}$ le codage standard des espaces de Banach séparables comme sous-espaces de $C\left(\Delta \right)$. Dans ce papier, on montre que si $𝔹\subset \text{SB}$ est un sous-ensemble borélien d’espaces à dual séparable, alors l’application $X\mapsto X^{*}$ peut être réalisée par une fonction borélienne de $𝔹$ à $\text{SB}$. En outre, cette application peut être construite de manière que l’évaluation fonctionnelle est toujours bien définie (Théorème 1). Par ailleurs, on démontre une version borélienne du théorème de Zippin. Plus précisément, on démontre qu’il existe $Z\in \text{SB}$ et une fonction borélienne qui à chaque $X$ associe une copie isomorphe à $X$ à l’intérieur de $Z$ (Théorème 5).

Let $\text{SB}$ be the standard coding for separable Banach spaces as subspaces of $C\left(\Delta \right)$. In these notes, we show that if $𝔹\subset \text{SB}$ is a Borel subset of spaces with separable dual, then the assignment $X\mapsto X^{*}$ can be realized by a Borel function $𝔹\to \text{SB}$. 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 $X\in 𝔹$ an isomorphic copy of $X$ inside of $Z$ (Theorem 5).

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