Some Characterizations of the Spaces L 1 (μ)

Abstract

Answers are given to the question of when the so-called hom and tensor functors in categories of Banach spaces preserve certain short exact sequences. The answers characterize the spaces of integrable, real-valued functions L1(u). Many questions can be raised concerning preservation properties of functors and their dual functors (see (8)) in categories of Banach spaces. Some are answered. For example, it can be shown if a functor preserves compact operators so does its dual functor. Similarly, we can ask the question: If a functor in categories of Banach spaces preserves certain types of exact sequences, does its dual functor do the same? Investigating this problem in particular for the hom and tensor functors, which are dual to each other, leads to the characterization of the spaces Ll(,u). The purpose of this paper is to give these characterizations.