Abstract
We formalize the observation that the same summability methods converge in a Banach space and its dual . At the same time we determine conditions under which these methods converge in weak and weak* topologies on and respectively. We also derive a general limitation theorem, which yields a necessary condition for the convergence of a summability method in . 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.
Acknowledgements
We thank the referees for their careful reading of the paper, and for correcting a mistake in an earlier version. We are also grateful to Ryan Gibara for helpful discussions.
Citation
Soumitra Ghara. Javad Mashreghi. Thomas Ransford. "SUMMABILITY AND DUALITY." Publ. Mat. 68 (2) 407 - 429, 2024. https://doi.org/10.5565/PUBLMAT6822403
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