Differentiation Theory over Infinite-Dimensional Banach Spaces

In this paper we study, for any positive integer $k$ and for any subset\ $I$\ of $\QTR{bf}{N}^{\ast }$, the Banach space $E_{I}$ of the bounded real sequences $\left\{ x_{n}\right\} _{n\in I}$, and a measure over $\left( \QTR{bf}{R}^{I},\QTR{cal}{B}^{(I)}\right) $ 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 $\left( \QTR{bf}{R}^{I},\QTR{cal}{B}^{(I)}\right) $. This change of variables is defined by some infinite-dimensional functions with properties that generalize the analogous ones of the standard finite-dimensional diffeomorphisms.