ABSTRACT FUNCTIONS OF BOUNDED VARIATION AND ABSOLUTE CONTINUITY

As well known that in 1938,I. M. Gelfand firstly introduced abstract functions of bounded variation from [a,b] to a Banach space. After Gelfand‘s work,many mathematicians investigated various properties and of this kind of abstract functions, and also paid attention to the abstract functions of absolute continuity In this paper, we summarize to explain our work [1-17] about this topic.     

A RIEMANN-TYPE DEFINITON OF THE BOCHNER INTEGRAL

We define the strong McShane integral and prove its equivalence with the Bochner integral In other words,we have given a Riemann-type definition of the Bochner integral.