Book contents
- Frontmatter
- Contents
- Introduction
- Part 1 Bohr’s Problem and Complex Analysis on Polydiscs
- Part 2 Advanced Toolbox
- Part 3 Replacing Polydiscs by Other Balls
- 18 Hardy–Littlewood Inequality
- 19 Bohr Radii in ℓp Spaces and Unconditionality
- 20 Monomial Convergence in Banach Sequence Spaces
- 21 Dineen’s Problem
- 22 Back to Bohr Radii
- Part 4 Vector-Valued Aspects
- References
- Symbol Index
- Subject Index
20 - Monomial Convergence in Banach Sequence Spaces
from Part 3 - Replacing Polydiscs by Other Balls
Published online by Cambridge University Press: 19 July 2019
- Frontmatter
- Contents
- Introduction
- Part 1 Bohr’s Problem and Complex Analysis on Polydiscs
- Part 2 Advanced Toolbox
- Part 3 Replacing Polydiscs by Other Balls
- 18 Hardy–Littlewood Inequality
- 19 Bohr Radii in ℓp Spaces and Unconditionality
- 20 Monomial Convergence in Banach Sequence Spaces
- 21 Dineen’s Problem
- 22 Back to Bohr Radii
- Part 4 Vector-Valued Aspects
- References
- Symbol Index
- Subject Index
Summary
The set of monomial convergence of the bounded holomophic functions on B_{c0} and of m-homogeneous polynomials on c0 was studied in Chapter 10. Here the space c0 is replaced by some other l_p spaces, or even by polynomials on an arbitrary Banach sequence space and holomorphic functions on Reinhardt domains. The only complete case is p=1, where the set of monomial convergence of the m-homogeneous polynomials is exactly l_1, and the set of monomial convergence of the bounded holomorphic functions on the open unit ball of l_1 is again the ball. For other p’s upper and lower bounds are presented that give a pretty tight description.
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- Information
- Dirichlet Series and Holomorphic Functions in High Dimensions , pp. 506 - 530Publisher: Cambridge University PressPrint publication year: 2019