A Generalized Riemann Integral for Banach-Valued Functions

Abstract

We shall develop the properties of an integral for Banach-valued functions. The formalism is the generalized Riemann integral introduced by Kurzweil \cite{Kur} and Henstock \cite{Hen}. More precisely, the presentation is close to the McShane approach \cite{McS}. Besides its simplicity of presentation, four advantages characterize this theory: %{\leftskip=1cm \item{(i)} the definition can be used for real-valued functions, and can be generalized without modification to general real and complex Banach spaces; \item{(ii)} when a function is integrable its norm is also integrable, and the proof is straightforward from the definition; \item{(iii)} for finite dimension spaces the theory is equivalent to the McShane's theory, which is itself equivalent to the Lebesgue's theory; \item{(iv)} and lastly, for general Banach space, we can prove the equivalence to the Bochner's theory. \par%}