Integration

Jacques Simon

ISBN 10: 1786300133 / ISBN 13: 9781786300133

Published by ISTE Ltd and John Wiley and Sons Inc, GB, 2026

About this Item

This book presents a simple and novel theory of integration, both real and vectorial, particularly suitable for the study of PDEs. This theory allows for integration with values in a Neumann space E, i.e. in which all Cauchy sequences converge, encompassing Neumann and Fréchet spaces, as well as "weak" spaces and distribution spaces. We integrate "integrable measures", which are equivalent to "classes of integrable functions which are a.e. equals" when E is a Fréchet space. More precisely, we associate the measure f with a class f, where f(u) is the integral of fu for any test function u. The classic space Lp(?;E) is the set of f, and ours is the set of f; these two spaces are isomorphic. Integration studies, in detail, for any Neumann space E, the properties of the integral and of Lp(?;E): regularization, image by a linear or multilinear application, change of variable, separation of multiple variables, compacts and duals. When E is a Fréchet space, we study the equivalence of the two definitions and the properties related to dominated convergence.

Seller Inventory # LU-9781786300133

Bibliographic Details

• Title: Integration
• Publisher: ISTE Ltd and John Wiley and Sons Inc, GB
• Publication Date: 2026
• Language: English
• ISBN 10: 1786300133
• ISBN 13: 9781786300133
• Binding: Hardback
• Condition: New

About this title

Synopsis

This book presents a simple and novel theory of integration, both real and vectorial, particularly suitable for the study of PDEs. This theory allows for integration with values in a Neumann space E, i.e. in which all Cauchy sequences converge, encompassing Neumann and Fréchet spaces, as well as "weak" spaces and distribution spaces.

We integrate "integrable measures", which are equivalent to "classes of integrable functions which are a.e. equals" when E is a Fréchet space. More precisely, we associate the measure f with a class f, where f(u) is the integral of fu for any test function u. The classic space Lp(Ω;E) is the set of f, and ours is the set of f; these two spaces are isomorphic.

Integration studies, in detail, for any Neumann space E, the properties of the integral and of Lp(Ω;E): regularization, image by a linear or multilinear application, change of variable, separation of multiple variables, compacts and duals. When E is a Fréchet space, we study the equivalence of the two definitions and the properties related to dominated convergence.

About the Author

Jacques Simon is Director Emeritus of Research at the CNRS, France. His research focuses on partial differential equations, particularly on the spaces used by these equations and on shape optimization.