On weak^*-extensible subspaces of Banach spaces

2021/03/05 by Gonzalo Martínez-Cervantes, Martínez-Cervantes, G., J. Rodríguez +1
Mathematics · #46B20 #46B50 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2103.03590

openalex publication_date 2021/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a Banach space and Y ⊆ X be a closed subspace. We prove that if the quotient X/Y is weakly Lindelöf determined or weak Asplund, then for every w^*-convergent sequence (y_n^*)_{n∈ \mathbb N} in Y^* there exist a subsequence (y_nk^*)_{k∈ \mathbb N} and a w^*-convergent sequence (x_k^*)_{k∈ \mathbb N} in X^* such that x_k^*|_Y=y_nk^* for all k∈ \mathbb N. As an application we obtain that Y is Grothendieck whenever X is Grothendieck and X/Y is reflexive, which answers a question raised by González and Kania.

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