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

Symbolverzeichnis......Page 0
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 Od......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