Book contents
- Frontmatter
- Contents
- Introduction
- 1 Complete Metric Spaces
- 2 Banach’s Principle
- 3 Picard’s Theorem
- 4 Banach Spaces
- 5 Renewal Equation in the McKendrick–von Foerster Model
- 6 Riemann Integral for Vector-Valued Functions
- 7 The Stone–Weierstrass Theorem
- 8 Norms Do Differ
- 9 Hilbert Spaces
- 10 Complete Orthonormal Sequences
- 11 Heat Equation
- 12 Completeness of the Space of Operators
- 13 Working in ℒ(𝕏)
- 14 The Banach–Steinhaus Theorem and Strong Convergence
- 15 We Go Deeper, DeeperWe Go (into the Structure of Complete Spaces)
- 16 Semigroups of Operators
- Appendix Two Consequences of the Hahn–Banach Theorem
- References
- Index
1 - Complete Metric Spaces
Published online by Cambridge University Press: 31 October 2024
Book contents
- Frontmatter
- Contents
- Introduction
- 1 Complete Metric Spaces
- 2 Banach’s Principle
- 3 Picard’s Theorem
- 4 Banach Spaces
- 5 Renewal Equation in the McKendrick–von Foerster Model
- 6 Riemann Integral for Vector-Valued Functions
- 7 The Stone–Weierstrass Theorem
- 8 Norms Do Differ
- 9 Hilbert Spaces
- 10 Complete Orthonormal Sequences
- 11 Heat Equation
- 12 Completeness of the Space of Operators
- 13 Working in ℒ(𝕏)
- 14 The Banach–Steinhaus Theorem and Strong Convergence
- 15 We Go Deeper, DeeperWe Go (into the Structure of Complete Spaces)
- 16 Semigroups of Operators
- Appendix Two Consequences of the Hahn–Banach Theorem
- References
- Index
Summary
Guided by basic intuitions, we introduce the notion of a complete metric space and discover that we have in fact encountered it before in our study of mathematics. In particular, we learn that if the set of real numbers were not complete, bounded increasing (or decreasing) sequences would not have limits. Similarly, we realize that if time were not complete, Achilles would never catch the tortoise. In a slightly more advanced part, we show that criteria for convergence of functional series involve the notion of completeness of the space of continuous functions.
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- Functional Analysis RevisitedAn Essay on Completeness, pp. 1 - 14Publisher: Cambridge University PressPrint publication year: 2024