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Aimed toward researchers and graduate students familiar with elements of functional analysis, linear algebra, and general topology; this book contains a general study of modulars, modular spaces, and metric modular spaces. Modulars may be thought of as generalized velocity fields and serve two important purposes: generate metric spaces in a unified manner and provide a weaker convergence, the modular convergence, whose topology is non-metrizable in general. Metric modular spaces are extensions of metric spaces, metric linear spaces, and classical modular linear spaces. The topics covered include the classification of modulars, metrizability of modular spaces, modular transforms and duality between modular spaces, metric and modular topologies. Applications illustrated in this book include: the description of superposition operators acting in modular spaces, the existence of regular selections of set-valued mappings, new interpretations of spaces of Lipschitzian and absolutely continuous mappings, the existence of solutions to ordinary differential equations in Banach spaces with rapidly varying right-hand sides.
List of contents
1.Classes of Modulars.- 2. Metrics on Modular Spaces.- 3. Modular Transforms.- 4. Topologies on Modular Spaces.- 5. Bounded and Regulated Mappings.- 6. Mappings of Bounded Generalized Variation.- Appendix.- References.
Summary
Aimed toward researchers and graduate students familiar with elements of functional analysis, linear algebra, and general topology; this book contains a general study of modulars, modular spaces, and metric modular spaces. Modulars may be thought of as generalized velocity fields and serve two important purposes: generate metric spaces in a unified manner and provide a weaker convergence, the modular convergence, whose topology is non-metrizable in general. Metric modular spaces are extensions of metric spaces, metric linear spaces, and classical modular linear spaces. The topics covered include the classification of modulars, metrizability of modular spaces, modular transforms and duality between modular spaces, metric and modular topologies. Applications illustrated in this book include: the description of superposition operators acting in modular spaces, the existence of regular selections of set-valued mappings, new interpretations of spaces of Lipschitzian and absolutely continuous mappings, the existence of solutions to ordinary differential equations in Banach spaces with rapidly varying right-hand sides.
Additional text
“Based extensively on the original results of the author, the book generalizes and presents in a unitary way a lot of results on classical modular spaces with applications to the study of function spaces and of nonlinear operators acting on them. It can be recommended to researchers in applied functional analysis, ordinary differential equations and partial differential equations.” (Stefan Cobzaş, zbMATH 1354.46001, 2017)
“This book is a contribution to the general theory of modulars defined on arbitrary sets. … The content of the book is well organized, and the exposition is as self-contained as possible. … this is an interesting book on function spaces theory. It contains a wealth of materials concerning modular spaces and some of their applications. This work should be useful to undergraduate students and certainly to a more general audience of mathematicians and physicists.” (Mohammed Morsli, Mathematical Reviews, September, 2016)
Report
"Based extensively on the original results of the author, the book generalizes and presents in a unitary way a lot of results on classical modular spaces with applications to the study of function spaces and of nonlinear operators acting on them. It can be recommended to researchers in applied functional analysis, ordinary differential equations and partial differential equations." (Stefan Cobzas, zbMATH 1354.46001, 2017)
"This book is a contribution to the general theory of modulars defined on arbitrary sets. ... The content of the book is well organized, and the exposition is as self-contained as possible. ... this is an interesting book on function spaces theory. It contains a wealth of materials concerning modular spaces and some of their applications. This work should be useful to undergraduate students and certainly to a more general audience of mathematicians and physicists." (Mohammed Morsli, Mathematical Reviews, September, 2016)
