The Radon-Nikod\acuteYm property of \mathbbL-Banach spaces and the dual representation theorem of \mathbbL-Bochner function spaces

2024/09/10 by Xia Zhang, Zhang, Xia, Ming Liu +2
Mathematics · #46B10 #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Primary 46B22 #Secondary 46E30

paper · pdf · doi:10.48550/arxiv.2409.06279

openalex publication_date 2024/09/10 · openalex created_date 2024/10/22 · openalex updated_date 2026/07/28

Abstract

In this paper, we first introduce \mathbbL-μ-measurable functions and \mathbbL-Bochner integrable functions on a finite measure space (S,F,μ), and give an \mathbbL-valued analogue of the canonical L^p(Ω,F,μ). Then we investigate the completeness of such an \mathbbL-valued analogue and propose the Radon-Nikod\acuteym property of \mathbbL-Banach spaces. Meanwhile, an example constructed in this paper shows that there do exist an \mathbbL-Banach space which fails to possess the Radon-Nikod\acuteym property. Finally, based on above work, we establish the dual representation theorem of \mathbbL-Bochner integrable function spaces, which extends and improves the corresponding classical result.

Related

• The near Radon-Nikodym property in Lebesgue-Bochner function spaces
• A Bourgain-Pisier construction for general Banach spaces
• Duality of Bochner spaces
• Radon-Nikodym property in L(X,Y)
• Compactness in the Lebesgue-Bochner spaces L^p(μ;X)