Differentiation Theory over Infinite-Dimensional Banach Spaces

Abstract

We study, for any positive integer k and for any subset I of N∗, the Banach space EI of the bounded real sequences {xn}n∈I and a measure over (RI,B(I)) that generalizes the k-dimensional Lebesgue one. Moreover, we expose a differentiation theory for the functions defined over this space. The main result of our paper is a change of variables' formula for the integration of the measurable real functions on (RI,B(I)). This change of variables is defined by some infinite-dimensional functions with properties that generalize the analogous ones of the standard finite-dimensional diffeomorphisms.