Open problems in Banach spaces and measure theory

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 σ-algebras, measurable selectors), vector integration (Riemann, Pettis and McShane integrals), vector measures (range and associated L¹ spaces) and Lebesgue-Bochner spaces (topological and structural properties, scalar convergence).

Mathematics Subject Classification (2010):

• Primary: 28B05
• 46G10
• Secondary: 28B20
• 46B26
• 46B50
• 46E30
• 46E40

Key words:

• Banach space
• Baire σ-algebra
• Borel σ-algebra
• measurable selector
• Riemann integral
• Pettis integral
• McShane integral
• vector measure
• Lebesgue-Bochner space