A Normed Space of Weakly Integrable Operator-Valued Functions and Convergence Theorems

Abstract

Let X and Y be Banach spaces. The first part of this paper deals with a normed space which consists of weakly integrable, in a suitable sense, -valued functions. We give another norm for which is equivalent to the initial one. We provide an example when this space is not a Banach space and we prove that it is a Banach space if the measure μ is discrete and Y is reflexive. We study some naturally defined operators on. In the second part we consider convergence theorems for sequences of functions in. Let At(n)t∈Ω be a sequence in and let (At)t∈Ω be a family in such that limn→∞At(n)=At for all t∈Ω, where the limit is in the weak, strong or uniform sense. Under some additional conditions we prove that (Formula presented.) where the limit is weak, strong and uniform respectively. These results generalize Dominant Convergence Theorem and Vitali Convergence Theorem. Moreover, a converse of the uniform version of Vitali Convergence Theorem is obtained.