Duality on Banach spaces and a Borel parametrized version of Zippin’s theorem
Annales de l'Institut Fourier, Volume 65 (2015) no. 6, p. 2413-2435

Let SB be the standard coding for separable Banach spaces as subspaces of C(Δ). In these notes, we show that if 𝔹SB is a Borel subset of spaces with separable dual, then the assignment XX * can be realized by a Borel function 𝔹SB. 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 ZSB and a Borel function that assigns for each X𝔹 an isomorphic copy of X inside of Z (Theorem 5).

Soit SB le codage standard des espaces de Banach séparables comme sous-espaces de C(Δ). Dans ce papier, on montre que si 𝔹SB est un sous-ensemble borélien d’espaces à dual séparable, alors l’application XX * peut être réalisée par une fonction borélienne de 𝔹 à 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 ZSB et une fonction borélienne qui à chaque X associe une copie isomorphe à X à l’intérieur de Z (Théorème 5).

DOI : https://doi.org/10.5802/aif.2991
Classification:  46B10
Keywords: Banach spaces, duality, descriptive set theory, Zippin’s theorem
@article{AIF_2015__65_6_2413_0,
     author = {Braga, Bruno de Mendon\c ca},
     title = {Duality on Banach spaces and a Borel parametrized version of Zippin's theorem},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {6},
     year = {2015},
     pages = {2413-2435},
     doi = {10.5802/aif.2991},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_6_2413_0}
}
Braga, Bruno de Mendonça. Duality on Banach spaces and a Borel parametrized version of Zippin’s theorem. Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2413-2435. doi : 10.5802/aif.2991. http://www.numdam.org/item/AIF_2015__65_6_2413_0/

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