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

Description

This is the second of three volumes that provide a full and detailed account of all those elements of real and complex analysis the undergraduate mathematics student can expect to encounter in their first two or three years of study. Numerous exercises, examples, and applications are included. Cover......Page 1 A COURSE IN MATHEMATICAL ANALYSIS......Page 3 Title......Page 5 Copyright......Page 6 Contents......Page 7 Introduction......Page 11 Part III Metric and topological spaces......Page 13 11.1 Metric spaces: examples......Page 15 11.2 Normed spaces......Page 21 11.3 Inner-product spaces......Page 24 11.4 Euclidean and unitary spaces......Page 29 11.5 Isometries......Page 31 11.6 *The Mazur–Ulam theorem*......Page 35 11.7 The orthogonal group bold0mu mumu OdOdOdOdOdOd......Page 39 12.1 Convergence of sequences in a metric space......Page 42 12.2 Convergence and continuity of mappings......Page 49 12.3 The topology of a metric space......Page 54 12.4 Topological properties of metric spaces......Page 61 13.1 Topological spaces......Page 65 13.2 The product topology......Page 73 13.3 Product metrics......Page 78 13.4 Separation properties......Page 82 13.5 Countability properties......Page 87 13.6 *Examples and counterexamples*......Page 91 14.1 Completeness......Page 98 14.2 Banach spaces......Page 107 14.3 Linear operators......Page 112 14.4 *Tietze's extension theorem*......Page 118 14.5 The completion of metric and normed spaces......Page 120 14.6 The contraction mapping theorem......Page 124 14.7 *Baire's category theorem*......Page 132 15.1 Compact topological spaces......Page 143 15.2 Sequentially compact topological spaces......Page 147 15.3 Totally bounded metric spaces......Page 151 15.4 Compact metric spaces......Page 153 15.5 Compact subsets of C(K)......Page 157 15.6 *The Hausdorff metric*......Page 160 15.7 Locally compact topological spaces......Page 164 15.8 Local uniform convergence......Page 169 15.9 Finite-dimensional normed spaces......Page 172 16.1 Connectedness......Page 176 16.2 Paths and tracks......Page 182 16.3 Path-connectedness......Page 185 16.4 *Hilbert's path*......Page 187 16.5 *More space-filling paths*......Page 190 16.6 Rectifiable paths......Page 192 Part IV Functions of a vector variable......Page 195 17.1 Differentiating functions of a vector variable......Page 197 17.2 The mean-value inequality......Page 203 17.3 Partial and directional derivatives......Page 208 17.4 The inverse mapping theorem......Page 212 17.5 The implicit function theorem......Page 214 17.6 Higher derivatives......Page 216 18.1 Elementary vector-valued integrals......Page 225 18.2 Integrating functions of several variables......Page 227 18.3 Integrating vector-valued functions......Page 229 18.4 Repeated integration......Page 237 18.5 Jordan content......Page 242 18.6 Linear change of variables......Page 246 18.7 Integrating functions on Euclidean space......Page 248 18.8 Change of variables......Page 249 18.9 Differentiation under the integral sign......Page 255 19.1 Differential manifolds in Euclidean space......Page 257 19.2 Tangent vectors......Page 260 19.3 One-dimensional differential manifolds......Page 264 19.4 Lagrange multipliers......Page 267 19.5 Smooth partitions of unity......Page 277 19.6 Integration over hypersurfaces......Page 280 19.7 The divergence theorem......Page 284 19.8 Harmonic functions......Page 294 19.9 Curl......Page 299 B.1 Finite-dimensional vector spaces......Page 303 B.2 Linear mappings and matrices......Page 306 B.3 Determinants......Page 309 B.4 Cramer's rule......Page 311 B.5 The trace......Page 312 C.1 Exterior algebras......Page 313 C.2 The cross product......Page 316 D Tychonoff's theorem......Page 319 Index......Page 324 Contents for Volume I......Page 330 Contents for Volume III......Page 333