A Course in Mathematical Analysis: Volume 2, Metric and Topological Spaces, Functions of a Vector Variable

Description

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 their 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 teachers. Volume I focuses on the analysis of real-valued functions of a real variable. This second volume goes on to consider metric and topological spaces. Topics such as completeness, compactness and connectedness are developed, with emphasis on their applications to analysis. This leads to the theory of functions of several variables. 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 theorem. Volume III covers complex analysis and the theory of measure and integration. Cover A COURSE IN MATHEMATICAL ANALYSIS A COURSE INMATHEMATICAL ANALYSIS, Volume II Copyright D. J. H. Garling 2013 ISBN 978-1-107-03203-3 ISBN 978-1-107-67532-2 QA611.28.G37 2013 514'.325–dc23 LCCN 2012044992 Contents Introduction Part III Metric and topological spaces 11 Metric spaces and normed spaces 11.1 Metric spaces: examples 11.2 Normed spaces 11.3 Inner-product spaces 11.4 Euclidean and unitary spaces 11.5 Isometries 11.6 *The Mazur–Ulam theorem* 11.7 The orthogonal group bold0mu mumu OdOdOdOdOdOd 12 Convergence, continuity and topology 12.1 Convergence of sequences in a metric space 12.2 Convergence and continuity of mappings 12.3 The topology of a metric space 12.4 Topological properties of metric spaces 13 Topological spaces 13.1 Topological spaces 13.2 The product topology 13.3 Product metrics 13.4 Separation properties 13.5 Countability properties 13.6 *Examples and counterexamples* 14 Completeness 14.1 Completeness 14.2 Banach spaces 14.3 Linear operators 14.4 *Tietze's extension theorem* 14.5 The completion of metric and normed spaces 14.6 The contraction mapping theorem 14.7 *Baire's category theorem* 15 Compactness 15.1 Compact topological spaces 15.2 Sequentially compact topological spaces 15.3 Totally bounded metric spaces 15.4 Compact metric spaces 15.5 Compact subsets of C(K) 15.6 *The Hausdorff metric* 15.7 Locally compact topological spaces 15.8 Local uniform convergence 15.9 Finite-dimensional normed spaces 16 Connectedness 16.1 Connectedness 16.2 Paths and tracks 16.3 Path-connectedness 16.4 *Hilbert's path* 16.5 *More space-filling paths* 16.6 Rectifiable paths Part IV Functions of a vector variable 17 Differentiating functions of a vector variable 17.1 Differentiating functions of a vector variable 17.2 The mean-value inequality 17.3 Partial and directional derivatives 17.4 The inverse mapping theorem 17.5 The implicit function theorem 17.6 Higher derivatives 18 Integrating functions of several variables 18.1 Elementary vector-valued integrals 18.2 Integrating functions of several variables 18.3 Integrating vector-valued functions 18.4 Repeated integration 18.5 Jordan content 18.6 Linear change of variables 18.7 Integrating functions on Euclidean space 18.8 Change of variables 18.9 Differentiation under the integral sign 19 Differential manifolds in Euclidean space 19.1 Differential manifolds in Euclidean space 19.2 Tangent vectors 19.3 One-dimensional differential manifolds 19.4 Lagrange multipliers 19.5 Smooth partitions of unity 19.6 Integration over hypersurfaces 19.7 The divergence theorem 19.8 Harmonic functions 19.9 Curl B Linear algebra B.1 Finite-dimensional vector spaces B.2 Linear mappings and matrices B.3 Determinants B.4 Cramer's rule B.5 The trace C Exterior algebras and the cross product C.1 Exterior algebras C.2 The cross product D Tychonoff's theorem Index Contents for Volume I Contents for Volume III