Book contents
- Frontmatter
- Contents
- Introduction
- 1 Complete Metric Spaces
- 2 Banachβs Principle
- 3 Picardβs Theorem
- 4 Banach Spaces
- 5 Renewal Equation in the McKendrickβvon Foerster Model
- 6 Riemann Integral for Vector-Valued Functions
- 7 The StoneβWeierstrass 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 β(π)
- 14 The BanachβSteinhaus 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 HahnβBanach Theorem
- References
- Index
13 - Working in β(π)
Published online by Cambridge University Press:Β 31 October 2024
Book contents
- Frontmatter
- Contents
- Introduction
- 1 Complete Metric Spaces
- 2 Banachβs Principle
- 3 Picardβs Theorem
- 4 Banach Spaces
- 5 Renewal Equation in the McKendrickβvon Foerster Model
- 6 Riemann Integral for Vector-Valued Functions
- 7 The StoneβWeierstrass 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 β(π)
- 14 The BanachβSteinhaus 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 HahnβBanach Theorem
- References
- Index
Summary
The space of bounded linear operators mapping a Banach space X into itself is not only a Banach space but also a Banach algebra with multiplication defined as composition. This provides additional possibilities of manipulation with elements of the space of operators. In particular, we can use `power seriesβ of operators to construct inverses of other operators, and thus solve linear equations in X. We can also define exponential functions of bounded linear operators to solve differential equations in X. Again, all of this would be impossible, were we not working in a complete space.
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- Information
- Functional Analysis RevisitedAn Essay on Completeness, pp. 144 - 162Publisher: Cambridge University PressPrint publication year: 2024