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.
Titolo: | Differentiation Theory over Infinite-Dimensional Banach Spaces | |
Autori: | ||
Data di pubblicazione: | 2016 | |
Stato di pubblicazione: | Pubblicato | |
Rivista: | ||
Abstract: | 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. | |
Handle: | http://hdl.handle.net/11368/2888977 | |
Digital Object Identifier (DOI): | http://dx.doi.org/10.1155/2016/2619087 | |
URL: | https://www.hindawi.com/journals/jmath/2016/2619087/abs | |
Appare nelle tipologie: | 1.1 Articolo in Rivista |
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