Metric Spaces: Iteration and ApplicationCambridge University Press, 02.05.1985 - 104 Seiten Here is an introductory text on metric spaces that is the first to be written for students who are as interested in the applications as in the theory. Knowledge of metric spaces is fundamental to understanding numerical methods (for example for solving differential equations) as well as analysis, yet most books at this level emphasise just the abstraction and theory. Dr Bryant uses applications to provide motivation and to sustain the development and discusses numerical procedures where appropriate. The reader is expected to have had some exposure to elementary analysis, but the author provides examples throughout to refresh the student's memory and to test and extend understanding. In short, this is an introductory textbook that will appeal to students of mathematics and engineering and will give them the required background for more advanced courses in both analysis and numerical analysis. |
Inhalt
Sequences by iteration | 1 |
13 Iterations in a different world | 7 |
Metric spaces | 12 |
22 Examples of metric spaces | 14 |
23 Sequences | 22 |
The three Cs | 29 |
32 Closed sets | 30 |
33 An internal test for convergence | 36 |
44 Some extensions | 64 |
45 Differential equations | 71 |
46 The implicit function theorem | 83 |
47 Conclusion | 86 |
What makes analysis work? | 87 |
52 Attained bounds | 92 |
53 Uniform continuity | 94 |
54 Inverse functions | 96 |
34 Complete sets | 38 |
35 Compact sets | 45 |
The contraction mapping principle | 52 |
42 Contractions | 57 |
43 Real contractions revisited | 62 |
55 Intermediate values | 98 |
56 Some final remarks | 103 |
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Häufige Begriffe und Wortgruppen
a₁ a₂ Cauchy sequence chapter closed and bounded closed sets compact set complete metric space continuity of f continuous functions contraction mapping principle convergent sequence convergent subsequence cubic cubic equation Deduce define d(x differential equation dx discrete space distance example exists a number f xk f¹(A finite number function f function given ƒ x2 ƒ x3 generalisation given by f(x graph of ƒ guess Hence i.e. there exist iterate of ƒ iterate with f Let f Let ƒ Let x1 limit n→oo open ball open sets point of f Proof Let proof of Theorem reader real analysis real function real numbers real sequence result sequence converges sequence x1 Show that ƒ space and let te[a Theorem 4.5 triangle inequality uniformly continuous unique fixed point unique solution usual metric verify x₁ x₁(t y₁ y₁)² ye X define เก
Verweise auf dieses Buch
An Introduction to Metric Spaces and Fixed Point Theory Mohamed A. Khamsi,William A. Kirk Eingeschränkte Leseprobe - 2001 |