A COURSE IN MATHEMATICAL ANALYSIS Volume II: Metric and Topological Spaces, Functions of a Vector Variable The three volumes of A Course in Mathematical Analysis provide a full and detailed account of all those elements of real and complex analysis that an undergraduate mathematics student can expect to encounter in the first two or three years of study. Containing hundreds of exercises, examples and applications, these books will become an invaluable resource for both students and instructors. Volume I focuses on the analysis of real-valued functions of a real vari- able. This second volume goes on to consider metric and topological spaces. Topics such as completeness, compactness and connectedness are devel- oped, with emphasis on their applications to analysis. This leads to the theory of functions of several variables: differentiation is developed in a coordinate free way, while integration (the Riemann integral) is established for functions defined on subsets of Euclidean space. Differential manifolds in Euclidean space are introduced in a final chapter, which includes an account of Lagrange multipliers and a detailed proof of the divergence the- orem. Volume III covers complex analysis and the theory of measure and integration. d. j. h. garling is Emeritus Reader in Mathematical Analysis at the University of Cambridge and Fellow of St. John’s College, Cambridge. He has fifty years’ experience of teaching undergraduate students in most areas of pure mathematics, but particularly in analysis. A COURSE IN MATHEMATICAL ANALYSIS Volume II Metric and Topological Spaces, Functions of a Vector Variable D. J. H. G A R L I N G Emeritus Reader in Mathematical Analysis, University of Cambridge, and Fellow of St John’s College, Cambridge TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStates ofAmericabyCambridgeUniversityPress,NewYork CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education, learningandresearchatthehighestinternational levelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781107675322 (cid:2)c D.J.H.Garling2013 Thispublicationisincopyright.Subjecttostatutory exception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2013 PrintedintheUnitedKingdombyCPIGroupLtd,CroydonCR04YY A catalogue record for this publication is available from the British Library Library of Congress Cataloguing inPublication Data Garling,D.J.H. Metricandtopologicalspaces,functions ofavector variable/D.J.H.Garling. pages cm.–(Acourseinmathematical analysis;volume2) Includesbibliographicalreferencesandindex. ISBN978-1-107-03203-3(hardback) –ISBN978-1-107-67532-2 (paperback) 1. Metricspaces 2. Topological spaces. 3. Vectorvaluedfunctions. I. Title. QA611.28.G37 2013 514(cid:2).325–dc23 2012044992 ISBN978-1-107-03203-3 Hardback ISBN978-1-107-67532-2Paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternal orthird-partyinternetwebsitesreferredtointhispublication, anddoesnotguaranteethatanycontent onsuchwebsites is,orwillremain, accurate orappropriate. Contents Volume II Introduction page ix Part Three Metric and topological spaces 301 11 Metric spaces and normed spaces 303 11.1 Metric spaces: examples 303 11.2 Normed spaces 309 11.3 Inner-product spaces 312 11.4 Euclidean and unitary spaces 317 11.5 Isometries 319 11.6 *The Mazur−Ulam theorem* 323 11.7 The orthogonal group O 327 d 12 Convergence, continuity and topology 330 12.1 Convergence of sequences in a metric space 330 12.2 Convergence and continuity of mappings 337 12.3 The topology of a metric space 342 12.4 Topological properties of metric spaces 349 13 Topological spaces 353 13.1 Topological spaces 353 13.2 The product topology 361 13.3 Product metrics 366 13.4 Separation properties 370 13.5 Countability properties 375 13.6 *Examples and counterexamples* 379 14 Completeness 386 14.1 Completeness 386 v vi Contents 14.2 Banach spaces 395 14.3 Linear operators 400 14.4 *Tietze’s extension theorem* 406 14.5 The completion of metric and normed spaces 408 14.6 The contraction mapping theorem 412 14.7 *Baire’s category theorem* 420 15 Compactness 431 15.1 Compact topological spaces 431 15.2 Sequentially compact topological spaces 435 15.3 Totally bounded metric spaces 439 15.4 Compact metric spaces 441 15.5 Compact subsets of C(K) 445 15.6 *The Hausdorff metric* 448 15.7 Locally compact topological spaces 452 15.8 Local uniform convergence 457 15.9 Finite-dimensional normed spaces 460 16 Connectedness 464 16.1 Connectedness 464 16.2 Paths and tracks 470 16.3 Path-connectedness 473 16.4 *Hilbert’s path* 475 16.5 *More space-filling paths* 478 16.6 Rectifiable paths 480 Part Four Functions of a vector variable 483 17 Differentiating functions of a vector variable 485 17.1 Differentiating functions of a vector variable 485 17.2 The mean-value inequality 491 17.3 Partial and directional derivatives 496 17.4 The inverse mapping theorem 500 17.5 The implicit function theorem 502 17.6 Higher derivatives 504 18 Integrating functions of several variables 513 18.1 Elementary vector-valued integrals 513 18.2 Integrating functions of several variables 515 18.3 Integrating vector-valued functions 517 18.4 Repeated integration 525 18.5 Jordan content 530 Contents vii 18.6 Linear change of variables 534 18.7 Integrating functions on Euclidean space 536 18.8 Change of variables 537 18.9 Differentiation under the integral sign 543 19 Differential manifolds in Euclidean space 545 19.1 Differential manifolds in Euclidean space 545 19.2 Tangent vectors 548 19.3 One-dimensional differential manifolds 552 19.4 Lagrange multipliers 555 19.5 Smooth partitions of unity 565 19.6 Integration over hypersurfaces 568 19.7 The divergence theorem 572 19.8 Harmonic functions 582 19.9 Curl 587 Appendix B Linear algebra 591 B.1 Finite-dimensional vector spaces 591 B.2 Linear mappings and matrices 594 B.3 Determinants 597 B.4 Cramer’s rule 599 B.5 The trace 600 Appendix C Exterior algebras and the cross product 601 C.1 Exterior algebras 601 C.2 The cross product 604 Appendix D Tychonoff’s theorem 607 Index 612 Contents for Volume I 618 Contents for Volume III 621