Open problems in Banach spaces and measure theory

TL;DR

This paper compiles open problems in Banach space theory, focusing on measure-theoretic aspects such as measurability, vector integration, vector measures, and Lebesgue-Bochner spaces, highlighting key unresolved questions.

Contribution

It systematically categorizes and presents open research questions in Banach spaces related to measure theory, serving as a guide for future investigations.

Findings

Identifies open problems in non-separable $L^p$ spaces and compactness.

Highlights unresolved issues in measurability and vector integration.

Points out gaps in understanding of Lebesgue-Bochner space properties.

Abstract

We collect several open questions in Banach spaces, mostly related to measure theoretic aspects of the theory. The problems are divided into five categories: miscellaneous problems in Banach spaces (non-separable $L^p$ spaces, compactness in Banach spaces, $w^*$-null sequences in dual spaces), measurability in Banach spaces (Baire and Borel $\sigma$-algebras, measurable selectors), vector integration (Riemann, Pettis and McShane integrals), vector measures (range and associated $L^1$ spaces) and Lebesgue-Bochner spaces (topological and structural properties, scalar convergence).