Jacques Simon - Continuous Functions

2 iii Continuous Functions Jacques Simon

3 iv First published 2020 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK www.iste.co.uk John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA www.wiley.com © ISTE Ltd 2020 The rights of Jacques Simon to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2020933955 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-010-2

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To Claire and Patricia, By your gaiety, “joie de vivre”, and femininity, you have embellished my life, and you have allowed me to conserve the tenacity needed for this endeavor

Analysis for PDEs Set

coordinated by

Jacques Blum

Jacques Simon

First published 2020 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd

27-37 St George’s Road

London SW19 4EU

UK

www.iste.co.uk

John Wiley & Sons, Inc.

111 River Street

Hoboken, NJ 07030

USA

www.wiley.com

© ISTE Ltd 2020

The rights of Jacques Simon to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2020933955

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-78630-010-2

Objective.This book is the second of six volumes in a series dedicated to the mathematical tools for solving partial differential equations derived from physics:

Volume 1: Banach, Frechet, Hilbert and Neumann Spaces;

Volume 2: Continuous Functions;

Volume 3: Distributions;

Volume 4: Lebesgue and Sobolev Spaces;

Volume 5: Traces;

Volume 6: Partial Differential Equations.

This second volume is devoted to the partial differentiation of functions and the construction of primitives, which is its inverse mapping, and to their properties, which will be useful for constructing distributions and studying partial differential equations later.

Target audience.We intended to find simple methods that require a minimal level of knowledge to make these tools accessible to the largest audience possible – PhD candidates, advanced students 1and engineers – without losing generality and even generalizing some standard results, which may be of interest to some researchers.

Originality.The construction of primitives, the Cauchy integral and the weighting with which they are obtained are performed for a function taking values in a Neumann space, that is, a space in which every Cauchy sequence converges.

Neumann spaces.The sequential completeness characterizing these spaces is the most general property of E that guarantees that the integral of a continuous function taking values in E will belong to it, see Case where E is not a Neumann space (§ 4.3, p. 92). This property is more general than the more commonly considered property of completeness, that is the convergence of all Cauchy filters; for example, if E is an infinite-dimensional Hilbert space, then E-weak is a Neumann space but is not complete [Vol. 1, Property (4.11), p. 82].

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