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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This book is the first of a set dedicated to the mathematical tools used in partial differential equations derived from physics. Its focus is on normed or semi-normed vector spaces, including the spaces of Banach, Fréchet and Hilbert, with new developments on Neumann spaces, but also on extractable spaces. The author presents the main properties of these spaces, which are useful for the construction of Lebesgue and Sobolev distributions with real or vector values and for solving partial differential equations. Differential calculus is also extended to semi-normed spaces. Simple methods, semi-norms, sequential properties and others are discussed, making these tools accessible to the greatest number of students - doctoral students, postgraduate students - engineers and researchers without restricting or generalizing the results.
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Veröffentlichungsjahr: 2017
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Table of Contents
Cover
Title
Copyright
Introduction
Familiarization with Semi-normed Spaces
Notations
Chapter 1: Prerequisites
1.1. Sets, mappings, orders
1.2. Countability
1.3. Construction of ℝ
1.4. Properties of ℝ
PART 1: SEMI-NORMED SPACES
Chapter 2: Semi-normed Spaces
2.1. Definition of semi-normed spaces
2.2. Convergent sequences
2.3. Bounded, open and closed sets
2.4. Interior, closure, balls and semi-balls
2.5. Density, separability
2.6. Compact sets
2.7. Connected and convex sets
Chapter 3: Comparison of Semi-normed Spaces
3.1. Equivalent families of semi-norms
3.2. Topological equalities and inclusions
3.3. Topological subspaces
3.4. Filtering families of semi-norms
3.5. Sums of sets
Chapter 4: Banach, Fréchet and Neumann Spaces
4.1. Metrizable spaces
4.2. Properties of sets in metrizable spaces
4.3. Banach, Fréchet and Neumann spaces
4.4. Compacts sets in Fréchet spaces
4.5. Properties of ℝ
4.6. Convergent sequences
4.7. Sequential completion of a semi-normed space
Chapter 5: Hilbert Spaces
5.1. Hilbert spaces
5.2. Projection in a Hilbert space
5.3. The space ℝ
d
Chapter 6: Product, Intersection, Sum and Quotient of Spaces
6.1. Product of semi-normed spaces
6.2. Product of a semi-normed space by itself
6.3. Intersection of semi-normed spaces
6.4. Sum of semi-normed spaces
6.5. Direct sum of semi-normed spaces
6.6. Quotient space
PART 2: CONTINUOUS MAPPINGS
Chapter 7: Continuous Mappings
7.1. Continuous mappings
7.2. Continuity and change of topology or restriction
7.3. Continuity of composite mappings
7.4. Continuous semi-norms
7.5. Continuous linear mappings
7.6. Continuous multilinear mappings
7.7. Some continuous mappings
Chapter 8: Images of Sets Under Continuous Mappings
8.1. Images of open and closed sets
8.2. Images of dense, separable and connected sets
8.3. Images of compact sets
8.4. Images under continuous linear mappings
8.5. Continuous mappings in compact sets
8.6. Continuous real mappings
8.7. Compacting mappings
Chapter 9: Properties of Mappings in Metrizable Spaces
9.1. Continuous mappings in metrizable spaces
9.2. Banach’s fixed point theorem
9.3. Baire’s theorem
9.4. Open mapping theorem
9.5. Banach–Schauder’s continuity theorem
9.6. Closed graph theorem
Chapter 10: Extension of Mappings, Equicontinuity
10.1. Extension of equalities by continuity
10.2. Continuous extension of mappings
10.3. Equicontinuous families of mappings
10.4. Banach–Steinhaus equicontinuity theorem
Chapter 11: Compactness in Mapping Spaces
11.1. The spaces (
X; F
) and (
X; F
)-pt
11.2. Zorn’s lemma
11.3. Compactness in (
X; F
)
11.4. An Ascoli compactness theorem in (
X; F
)-pt
Chapter 12: Spaces of Linear or Multilinear Mappings
12.1. The space (
E; F
)
12.2. Bounded sets in (
E; F
)
12.3. Sequential completeness of (
E; F
) when
E
is metrizable
12.4. Semi-norms and norm on (
E; F
) when
E
is normed
12.5. Continuity of the composition of linear mappings
12.6. Inversibility in the neighborhood of an isomorphism
12.7. The space
d
(
E
1
×· · ·×
E
d
;
F
)
12.8. Separation of the variables of a multilinear mapping
PART 3: WEAK TOPOLOGIES
Chapter 13: Duality
13.1. Dual
13.2. Dual of ametrizableor normed space
13.3. Dual of a Hilbert space
13.4. Extraction of ∗ weakly converging subsequences
13.5. Continuity of the bilinear form of duality
13.6. Dual of a product
13.7. Dual of a direct sum
Chapter 14: Dual of a Subspace
14.1. Hahn–Banach theorem
14.2. Corollaries of the Hahn–Banach theorem
14.3. Characterization of a dense subspace
14.4. Dual of a subspace
14.5. Dual of an intersection
14.6. Dangerous identifications
Chapter 15: Weak Topology
15.1. Weak topology
15.2. Weak continuity and topological inclusions
15.3. Weak topology of a product
15.4. Weak topology of an intersection
15.5. Norm and semi-norms of a weak limit
Chapter 16: Properties of Sets for the Weak Topology
16.1. Banach–Mackey theorem (weakly bounded sets)
16.2. Gauge of a convex open set
16.3. Mazur’s theorem (weakly closed convex sets)
16.4. Šmulian’s theorem (weakly compact sets)
16.5. Semi-weak continuity of a bilinear mapping
Chapter 17: Reflexivity
17.1. Reflexive spaces
17.2. Sequential completion of a semi-reflexive space
17.3. Prereflexivity of metrizable spaces
17.4. Reflexivity of Hilbert spaces
17.5. Reflexivity of uniformly convex Banach spaces
17.6. A property of the combinations of linear forms
17.7. Characterizations of semi-reflexivity
17.8. Reflexivity of a subspace
17.9. Reflexivity of the image of a space
17.10. Reflexivity of the dual
Chapter 18: Extractable Spaces
18.1. Extractable spaces
18.2. Extractability of Hilbert spaces
18.3. Extractability of semi-reflexive spaces
18.4. Extractability of a subspace or of the image of a space
18.5. Extractability of a product or of a sum of spaces
18.6. Extractability of an intersection of spaces
18.7. Sequential completion of extractable spaces
PART 4: DIFFERENTIAL CALCULUS
Chapter 19: Differentiable Mappings
19.1. Differentiable mappings
19.2. Differentiality, continuity and linearity
19.3. Differentiation and change of topology or restriction
19.4. Mean value theorem
19.5. Bounds on a real differentiable mapping
19.6. Differentiation of a composite mapping
19.7. Differential of an inverse mapping
19.8. Inverse mapping theorem
Chapter 20: Differentiation of Multivariable Mappings
20.1. Partial differentiation
20.2. Differentiation of a multilinear or multi-component mapping
20.3. Differentiation of a composite multilinear mapping
Chapter 21: Successive Differentiations
21.1. Successive differentiations
21.2. Schwarz’s symmetry principle
21.3. Successive differentiations of a composite mapping
Chapter 22: Derivation of Functions of One Real Variable
22.1. Derivative of a function of one real variable
22.2. Derivative of a real function of one real variable
22.3. Leibniz formula
22.4. Derivatives of the power, logarithm and exponential functions
Bibliography
Cited Authors
Index
End User License Agreement
Landmarks
Cover
Table of Contents
Begin Reading
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e1
“Analysis is sometimes a means of becoming disgusted piecemeal with something that as a whole was bearable.”
PAUL VALERY Bad Thoughts and Others
“For the Nation, Science and Glory.”
NAPOLEON BONAPARTE (Motto for the Ecole Polytechnique)
Analysis for PDEs Setcoordinated by Jacques Blum
Volume 1
Banach, Fréchet, Hilbert and Neumann Spaces
Jacques Simon
First published 2017 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 Ltd27-37 St George’s RoadLondon SW19 4EUUKwww.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USAwww.wiley.com
© ISTE Ltd 2017The 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: 2017932960
British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-009-6
Introduction
Purpose. This book is the first of a set of books dedicated to the mathematical tools for partial derivative equations existing in physics.
This first volume is dedicated to normed or semi-normed vector spaces, including Banach, Fréchet and Hilbert spaces, with new developments concerning Neumann spaces ― which designate those in which every Cauchy sequence converges ― and extractable spaces ― those in which every bounded sequence has a weakly convergent subsequence.
The book presents the main properties of these spaces employed for the construction of distributions, Lebesgue and Sobolev spaces and traces, with real or Neumann space values, that are the subject of Volumes II [102], III [103] and IV [104], respectively, and in the resolution of partial differential equations, in Volume V [105].
To this end, differential calculus is extended to semi-normed spaces in the present Volume I.
Target audience. We have looked for simple methods requiring a minimal level of knowledge in order to make these tools accessible to the largest possible audience ― doctoral students, PhD students, engineers ― without restricting their generality and even by generalizing some results, so that this book is also intended for researchers. This has led us to a non-conventional approach that emphasizes semi-norms and sequential properties, whether regarding completeness, compactness or continuity.
Why semi-normed spaces? We do not restrict ourselves to normed spaces because essential spaces for PDEs are not normable, such as -weak.
We are interested in vector spaces E endowed with a family of semi-norms, rather than with a locally convex topology which is equivalent, in order to define the differentiability (p. 275) therein by comparing the semi-norms of a variation in the data to the semi-norms of the variation in the mapping. This also allows us to defineLp(Ω; E) (Volume III), because a semi-norm can be raised to a power p, not a neighborhood!
In addition, the study is easier with semi-norms than with the topology, although less usual: it follows that of normed spaces, the main difference consisting of working on several semi-norms or norms instead of a single norm.
Neumann spaces. We particularly focus on the spaces that are sequentially complete ― we call them Neumann spaces ― because it is essential that E has this property to define ′(Ω; E) and Lp(Ω; E): it is the one that guarantees them satisfactory properties and especially that when and that , as will be seen in Volume II. The study of these spaces is a step in that of spaces useful for evolution PDEs such as , or .
Despite the name sequentially complete being less simple, the concept itself is simpler than complete (see Definition 4.8, p. 55) and, above all, more general: for example, if E is a Hilbert space with infinite dimension, E-weak is sequentially complete but is not complete (see (4.11), p. 63).
Extractable and reflexive spaces. We exhaustively study spaces in which any bounded sequence has a weakly convergent subsequence ― we call them extractable ― because numerous results of existence of solutions for PDEs are based on this property.
We also study reflexivity in detail because it provides extractability results. We balance and simplify its definition by introducing a new notion, pre-reflexivity (Definition 17.1, p. 243).
Other sequential properties. We also insist on:
―
Sequential compactness
because it is the basis for extractability: it ensures that every sequence has a convergent subsequence, which compactness does not ensure (see (2.6), p. 27).
―
Sequential density
that provides better approximations than density.
―
Sequential continuity
because important mappings are sequentially continuous but not continuous, such as the duality bilinear form of a non-normable space (Theorem 13.22, p. 204) or the composition of linear mappings (Theorem 12.12, p. 171).
Moreover, we do not overlook topological properties because they are distinct from their sequential counterparts in non-metrizable spaces.
Weak topologies. The topology of E-weak is very simply defined by the semi-norms indexed by . Similarly, the topology of E′-∗weak is defined by indexed by .
Differential calculus. We study differentiability in any separated semi-normed space E because it is an important tool for the study of distributions with values in such a space. It seemed that it was only achieved in a normed space or in particular cases such as scalar differentiation that corresponds to differentiation in E-weak.
Numerous conventional properties remain but not all. For example, we are not able to prove the second- or higher-order differentiability of a composite mapping when the intermediary space is not normed (see Theorem 21.6, p. 308) or the continuity of the differential of a composite of continuously differentiable mappings (see Theorem 19.18, p. 287).
Novelties. The extension of differential calculus to separated semi-normed spaces (Chapters 19 to 21) appears as new to us. The same happens with the characterizations of the dual and of the weak topology of an intersection of semi-normed spaces (Theorems 14.9, p. 215 and 15.11, p. 227), and with the extractability property of such an intersection (Theorem 18.14, p. 271). We can also include the introduction of Neumann spaces (Definition 4.10, p. 55), extractable spaces (Definition 18.1, p. 265), pre-reflexivity (Definition 17.1, p. 243) and compactant mappings (Definition 8.14, p. 125).
The pre-reflexivity of metrizable spaces (Theorem 17.8, p. 249) and, more generally, of infra-barreled spaces (Theorem 17.11, p. 250) is new in this form (which is equivalent to previous results, see Note 4, p. 248).
Prerequisites. The proofs in the body of text are not based on any external result, with the exception of countability and ℝ properties whose statements are recalled in Chapter 1. Indeed, it seemed interesting to recall all of the necessary knowledge given the unorthodox course of this book: the study of semi-normed spaces without resorting to the general topology, distributions construction (in Volume II) before that of integrable functions (in Volume III), etc.
Comments. Unlike the body of the text, the comments written in small characters may make use of external or not yet established results. The first chapter, 1 Prerequisites, also appears in small print because it could also be admitted or omitted, as well as the last section, 22.4 Derivatives of the power, logarithm and exponential functions.
Reminders. This book is written such that it can be read in an out-of-order fashion by a non-expert: the proofs are detailed including arguments that may be trivial for an expert and the numbers of the theorems being used are systematically recalled. The author asks for the reader’s indulgence regarding the heaviness that may result thereof.
History. The origin of the concepts and of the results is specified as much as possible, in footnotes, hoping that the unavoidable injustices will be forgiven and, above all, reported to the author (for re-editions!). In addition to the opportunity offered to state what is believed to be new, history shows that behind every theorem there is a man or woman, sometimes contemporary, sometimes ancient and wise, that our distant ancestors ― Greeks included ― reasoned as well as us and that simple ideas yet remain(ed) to be found.
Navigation through this book:
― The
table of contents
, at the beginning of the book, gives the list of the topics addressed.
― The
index
, p. 335, provides another thematic access.
― The
notation table
, p. xvii, clarifies their meaning when doubt remains.
― Assumptions are all stated inside the theorems themselves.
Acknowledgments. Enrique FERNÁNDEZ-CARA suggested to me a large number of pertinent improvements. Olivier BESSON and Fulbert MIGNOT have also contributed to substantial improvements. The manuscript and the proofs have been carefully re-read by Jérôme LEMOINE and Pierre DREYFUSS; they are thus entirely responsible for any error that may subsist. Jacques BLUM has the great merit of having snatched the manuscript from its torpor.
Donald KNUTH has graciously provided the scientific community with his remarkable TEX software, utilized to compose this book.
I am extremely pleased to thank them.
Jacques SIMON Chapdes-Beaufort March 2017
