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
4 - Banach 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
We are finally introduced to the fundamental notion of functional analysis: the Banach space, a unique blend of notions of linear algebra and metric topology. We get to know a number of classical, elementary Banach spaces. Also, examples of normed linear spaces that are not complete teach us that in a Banach space its `extent’ and its norm match each other tightly.
Keywords
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- Chapter
- Information
- Functional Analysis RevisitedAn Essay on Completeness, pp. 37 - 54Publisher: Cambridge University PressPrint publication year: 2024