Contents
IX
Chapter 9. Developments and Limitations of the Riemann Integral 9 195
§9.1. Why go further? 195
§9.2. Improper integrals 9 197
§9.3. Integrals over areas 9 201
§9.4. The Riemann-Stieltjes integral 9 206
§9.5. How long is a piece of string? 9 213
Chapter 10. Metric Spaces 221
§10.1. Sphere packing 9 221
§10.2. Shannon's theorem V 224
§10.3. Metric spaces 229
§10.4. Norms and the interaction of algebra and analysis 234
§10.5. Geodesies 9 241
Chapter 11. Complete Metric Spaces 249
§11.1. Completeness 249
§11.2. The Bolzano-Weierstrass property 257
§11.3. The uniform norm 261
§11.4. Uniform convergence 265
§11.5. Power series 273
§11.6. Fourier series 9 282
Chapter 12. Contraction Mappings and Differential Equations 287
§12.1. Banach's contraction mapping theorem 287
§12.2. Existence of solutions of differential equations 289
§12.3. Local to global 9 294
§12.4. Green's function solutions 9 301
Chapter 13. Inverse and Implicit Functions 311
§13.1. The inverse function theorem 311
§13.2. The implicit function theorem 9 320
§13.3. Lagrange multipliers 9 328
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