An Introduction to Functional Analysis.pdf

1、An Introduction to Functional Analysis This accessible text covers key results in functional analysis that are essential for fur-ther study in the calculus of variations,analysis,dynamical systems,and the theory of partial differential equations.The treatment of Hilbert spaces covers the topics requ

2、ired to prove the Hilbert-Schmidt Theorem,including orthonormal bases,the Riesz Repre-sentation Theorem,and the basics of spectral theory.The material on Banach spaces and their duals includes the Hahn-Banach Theorem,the Krein-Milman Theorem,and results based on the Baire Category Theorem,before cul

3、minating in a proof of sequen-tial weak compactness in reflexive spaces.Arguments are presented in detail,and more than 200 fully-worked exercises are included to provide practice applying techniques and ideas beyond the major theorems.Familiarity with the basic theory of vector spaces and point-set

4、 topology is assumed,but knowledge of measure theory is not required,making this book ideal for upper undergraduate-level and beginning graduate-level courses.JAME s Ro BIN so N is a professor in the Mathematics Institute at the University of Warwick.He has been the recipient of a Royal Society Univ

5、ersity Research Fellowship and an EPSRC Leadership Fellowship.He has written six books in addition to his many publications in infinite-dimensional dynamical systems,dimension theory,and partial differential equations.An Introduction to Functional Analysis JAMES C.ROBINSON University of Warwick CAMB

6、RIDGE UNIVERSITY PRESS CAMBRIDGE UNIVERSITY PRESS University Printing House,Cambridge CB2 8BS,United Kingdom One Liberty Plaza,20th Floor,New York,NY 10006,USA 477 Williamstown Road,Port Melbourne,VIC 3207,Australia 314-321,3rd Floor,Plot 3,Splendor Forum,Jasola District Centre,New Delhi-110025,Indi

7、a 79 Anson Road,#06-04/06,Singapore 079906 Cambridge University Press is part of the University of Cambridge.It furthers the Universitys mission by disseminating knowledge in the pursuit of education,learning,and research at the highest international levels of excellence.www.cambridge.org Informatio

8、n on this title:www.cambridge.org/9780521899642 DOI:10.1017/9781139030267 James C.Robinson 2020 This publication is in copyright.Subject to statutory exception and to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the written permission

9、of Cambridge University Press.First published 2020 Printed in the United Kingdom by TJ International Ltd.Padstow Cornwall A catalogue record for this publication is available from the British Library.ISBN 978-0-521-89964-2 Hardback ISBN 978-0-521-72839-3 Paperback Cambridge University Press has no r

10、esponsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is,or will remain,accurate or appropriate.ToMum&Dad 1 2 3 Contents Preface PART I PRELIMINARIES Vector Spaces a

11、nd Bases 1.1 Definition of a Vector Space 1.2 Examples of Vector Spaces 1.3 Linear Subspaces page xm 1 3 3 4 6 1.4 Spanning Sets,Linear Independence,and Bases 7 1.5 Linear Maps between Vector Spaces and Their Inverses 10 1.6 Existence of Bases and Zorns Lemma 13 Exercises Metric Spaces 2.1 Metric Sp

12、aces 2.2 Open and Closed Sets 2.3 Continuity and Sequential Continuity 2.4 Interior,Closure,Density,and Separability 2.5 Compactness Exercises PART II NORMED LINEAR SPACES Norms and Normed Spaces 3.1 Norms 3.2 Examples of Normed Spaces 3.3 Convergence in Normed Spaces 3.4 Equivalent Norms 3.5 Isomor

13、phisms between Normed Spaces 3.6 Separability of Normed Spaces Exercises Vll 15 17 17 19 22 23 25 30 33 35 35 38 42 43 46 48 50 Vlll Contents 4 Complete Normed Spaces 53 4.1 Banach Spaces 53 4.2 Examples of Banach Spaces 56 4.2.1 Sequence Spaces 57 4.2.2 Spaces of Functions 58 4.3 Sequences in Banac

14、h Spaces 61 4.4 The Contraction Mapping Theorem 63 Exercises 64 5 Finite-Dimensional Normed Spaces 66 5.1 Equivalence of Norms on Finite-Dimensional Spaces 66 5.2 Compactness of the Closed Unit Ball 68 Exercises 70 6 Spaces of Continuous Functions 71 6.1 The Weierstrass Approximation Theorem 71 6.2

15、The Stone-Weierstrass Theorem 77 6.3 The Arzela-Ascoli Theorem 83 Exercises 86 7 Completions and the Lebesgue Spaces LP(Q)89 7.1 Non-completeness of C(O,l)with the L1 Norm 89 7.2 The Completion of a Normed Space 91 7.3 Definition of the LP Spaces as Completions 94 Exercises 97 PART III HILBERT SPACE

16、S 99 8 Hilbert Spaces 101 8.1 Inner Products 101 8.2 The Cauchy-Schwarz Inequality 103 8.3 Properties of the Induced Norms 105 8.4 Hilbert Spaces 107 Exercises 108 9 Orthonormal Sets and Orthonormal Bases for Hilbert Spaces 110 9.1 Schauder Bases in Normed Spaces 110 9.2 Orthonormal Sets 112 9.3 Con

17、vergence of Orthogonal Series 115 9.4 Orthonormal Bases for Hilbert Spaces 117 9.5 Separable Hilbert Spaces 122 Exercises 123 Contents IX 10 Closest Points and Approximation 126 10.1 Closest Points in Convex Subsets of Hilbert Spaces 126 10.2 Linear Subspaces and Orthogonal Complements 129 10.3 Best

18、 Approximations 131 Exercises 134 11 Linear Maps between Normed Spaces 137 11.1 Bounded Linear Maps 137 11.2 Some Examples of Bounded Linear Maps 141 11.3 Completeness of B(X,Y)When Y Is Complete 145 11.4 Kernel and Range 146 11.5 Inverses and Invertibility 147 Exercises 150 12 Dual Spaces and the R

19、iesz Representation Theorem 153 12.1 The Dual Space 153 12.2 The Riesz Representation Theorem 155 Exercises 157 13 The Hilbert Adjoint of a Linear Operator 159 13.1 Existence of the Hilbert Adjoint 159 13.2 Some Examples of the Hilbert Adjoint 162 Exercises 164 14 The Spectrum of a Bounded Linear Op

20、erator 165 14.1 The Resolvent and Spectrum 165 14.2 The Spectral Mapping Theorem for Polynomials 169 Exercises 171 15 Compact Linear Operators 173 15.1 Compact Operators 173 15.2 Examples of Compact Operators 175 15.3 Two Results for Compact Operators 177 Exercises 178 16 The Hilbert-Schmidt Theorem

21、 180 16.1 Eigenvalues of Self-Adjoint Operators 180 16.2 Eigenvalues of Compact Self-Adjoint Operators 182 16.3 The Hilbert-Schmidt Theorem 184 Exercises 188 17 Application:Sturm-Liouville Problems 190 17.1 Symmetry of Land the Wronskian 191 x Contents 17.2 The Greens Function 193 17.3 Eigenvalues o

22、f the Sturm-Liouville Problem 195 PART IV BANACH SPACES 199 18 Dual Spaces of Banach Spaces 201 18.1 The Young and Holder Inequalities 202 18.2 The Dual Spaces of,f,P 204 18.3 Dual Spaces of LP(Q)207 Exercises 208 19 The Hahn-Banach Theorem 210 19.1 The Hahn-Banach Theorem:Real Case 210 19.2 The Hah

23、n-Banach Theorem:Complex Case 214 Exercises 217 20 Some Applications of the Hahn-Banach Theorem 219 20.1 Existence of a Support Functional 219 20.2 The Distance Functional 220 20.3 Separability of X*Implies Separability of X 221 20.4 Ad joints of Linear Maps between Banach Spaces 222 20.5 Generalise

24、d Banach Limits 224 Exercises 226 21 Convex Subsets of Banach Spaces 228 21.1 The Minkowski Functional 228 21.2 Separating Convex Sets 230 21.3 Linear Functionals and Hyperplanes 233 21.4 Characterisation of Closed Convex Sets 234 21.5 The Convex Hull 235 21.6 The Krein-Milman Theorem 236 Exercises

25、239 22 The Principle of Uniform Boundedness 240 22.1 The Baire Category Theorem 240 22.2 The Principle of Uniform Boundedness 242 22.3 Fourier Series of Continuous Functions 244 Exercises 247 23 The Open Mapping,Inverse Mapping,and Closed Graph Theorems 249 23.1 The Open Mapping and Inverse Mapping

26、Theorems 249 23.2 Schauder Bases in Separable Banach Spaces 252 Contents 23.3 The Closed Graph Theorem Exercises 24 Spectral Theory for Compact Operators 24.1 Properties of T-I When T Is Compact 24.2 Properties of Eigenvalues Xl 255 256 258 258 262 25 Unbounded Operators on Hilbert Spaces 264 25.1 A

27、djoints of Unbounded Operators 265 25.2 Closed Operators and the Closure of Symmetric Operators 267 25.3 The Spectrum of Closed Unbounded Self-Adjoint Operators 269 26 Reflexive Spaces 273 26.1 The Second Dual 273 26.2 Some Examples of Reflexive Spaces 26.3 X Is Reflexive If and Only If X*Is Reflexi

28、ve Exercises 27 Weak and Weak-*Convergence 27.1 Weak Convergence 275 277 280 282 282 27.2 Examples of Weak Convergence in Various Spaces 285 27.2.1 Weak Convergence in f,P,1 p.x E V for.E IK,x E V,(1.1)(i)additive and multiplicative identities exist:there exists a zero element 0 E V such that x+0=x

29、for all x E V;and 1 E IK is the identity for scalar multiplication,Ix=x for all x E V;(ii)there are additive inverses:for every x E V there exists an element-x E V such that x+(-x)=0;3 4 Vector Spaces and Bases(iii)addition is commutative and associative,and for all x,y,z E V;and x+(y+z)=(x+y)+z,(iv

30、)multiplication is associative,a(f3x)=(af3)x and distributive,a(x+y)=ax+ay for all a,f3 E IK,x,y E V.for all Ci,f3 E IK,X E V,and(a+f3)x=ax+f3x In checking that a particular collection V is a vector space over IK,properties(i)-(iv)are often immediate;one usually has to check only that V is closed un

31、der addition and scalar multiplication(i.e.that(1.1)holds).1.2 Examples of Vector Spaces Of course,IR.n is a real vector space over IR;but is not a vector space over C,since ix IRn for any1 x E IRn.In contrast,ccn can be a vector space over both IR and C;the space ccn over IR is(according to the ter

32、minology intro-duced above)a real vector space.This example is a useful illustration that the real/complex label refers to the field IK,i.e.the allowable scalar multiples,rather than to the elements of the space itself.Given any two vector spaces Vi and V2 over IK,the product space Vi x V2 consistin

33、g of all pairs(v1,v2)with v1 E Vi and v2 E V2 is another vector space if we define and for v1,u1 E Vi,v2,u2 E V2,a E IK.We now introduce some less trivial examples.Example 1.2 The space J(U,V)of all functions f:U-+V,where U and V are both vector spaces over the same field IK,is itself a vector space,if we use the obvious definitions of what addition and scalar multiplication should mean for functi