Textbook

Index i-vi

Chapter 1. Distance and size

Section 1. Metric spaces I-2
Subsection 1.1 Examples of metric spaces I-4
Subsection 1.2 Equivalence of metrics I-21
Subsection 1.3 Holder and Minkowsky inequalities I-30
Subsection 1.4 Lebesgue spaces I-43

Section 2. Normed linear spaces I-56
Subsection 2.1 Linear spaces I-57
Subsection 2.2 Norms I-60
Subsection 2.3 Application: norms of matrices I-63
Subsection 2.4 Application: differential equations I-84

Section 3. Topology I-106
Subsection 3.1 Metric space topology I-109
Subsection 3.2 Continuity I-119
Subsection 3.3 Hausdorff spaces I-131
Subsection 3.4 How to define topologies I-136
Subsection 3.5 Vocabulary of topology I-144
Subsection 3.6 Application: Brownian motion I-151
Subsection 3.7 Application: weak topology I-157

Section 4. Application: theory of distributions I-171
Subsection 4.1 Examples of distributions I-177
Subsection 4.2 Test function spaces I-198
Subsection 4.3 Generalized functions I-217

Section 5. The size of sets I-227
Subsection 5.1 Small sets, large sets I-228
Subsection 5.2 Measure theory I-239
Subsection 5.3 Application: Hausdorff measure I-252
Subsection 5.4 Application: Brownian motion I-264

Chapter 2. Convergence and compactness

Section 1. Convergence and compactness Il-282
Subsection 1.1 Cauchy sequences II-284
Subsection 1.2 Completions Il-299
Subsection 1.3 Compactness II-303
Subsection 1.4 Uniformity Il-314

Section 2. The Lebesgue integral Il-334
Subsection 2.1 Completion of C([0, 1]) II-335
Subsection 2.2 On defining integrals Il-338
Subsection 2.3 Measurable functions II-347
Subsection 2.4 The Lebesgue integral II-353
Subsection 2.5 Convergence theorems Il-361
Subsection 2.6 The Lebesgue spaces II-373
Subsection 2.7 Fubini's theorem II-387

Section 3. Arzela-Ascoli theorem II-394
Subsection 3.1 Solutions of ordinary differential equations II-398
Subsection 3.2 The length of curves II-403