Summability and duality
We formalize the observation that the same summability methods converge in a Banach space X and its dual X∗. At the same time we determine conditions under which these methods converge in weak and weak* topologies on X and X∗ respectively. We also derive a general limitation theorem, which yields a...
| Authors: | , , |
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| Format: | article |
| Publication Date: | 2024 |
| Country: | España |
| Institution: | Universitat Autònoma de Barcelona |
| Repository: | Dipòsit Digital de Documents de la UAB |
| Language: | English |
| OAI Identifier: | oai:ddd.uab.cat:294998 |
| Online Access: | https://ddd.uab.cat/record/294998 |
| Access Level: | Open access |
| Keyword: | Summability Limitation theorem Cesàro mean Banach space Dual space |
| Summary: | We formalize the observation that the same summability methods converge in a Banach space X and its dual X∗. At the same time we determine conditions under which these methods converge in weak and weak* topologies on X and X∗ respectively. We also derive a general limitation theorem, which yields a necessary condition for the convergence of a summability method in X. These results are then illustrated by applications to a wide variety of function spaces, including spaces of continuous functions, Lebesgue spaces, the disk algebra, Hardy and Bergman spaces, the BMOA space, the Bloch space, and de Branges-Rovnyak spaces. Our approach shows that all these applications flow from just two abstract theorems. |
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