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
6 - Riemann Integral for Vector-Valued Functions
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
The Riemann integral for vector-valued functions can be defined in the same way as for scalar-valued functions. Moreover, the theory of the so-defined integral is rather similar to the classical one. In particular, any continuous function with values in a Banach space is Riemann integrable, and the fundamental theorem of calculus remains valid. The theory of Riemann integration for functions with values in a normed, not complete, space would not be so elegant.
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- Information
- Functional Analysis RevisitedAn Essay on Completeness, pp. 65 - 69Publisher: Cambridge University PressPrint publication year: 2024