Differentiation theory over infinite-dimensional Banach spaces (Q670554)

Summary: We study, for any positive integer \(k\) and for any subset \(I\) of \(\mathbb{N}^\ast\), the Banach space \(E_I\) of the bounded real sequences \(\{x_n\}_{n \in I}\) and a measure over \((\mathbb{R}^I, \mathcal B^{(I)})\) 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 \((\mathbb{R}^I, \mathcal B^{(I)})\). This change of variables is defined by some infinite-dimensional functions with properties that generalize the analogous ones of the standard finite-dimensional diffeomorphisms.