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Note on almost isometric ideals and local retracts in Banach and metric spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices Model theory

Published online by Cambridge University Press:  27 August 2024

Leandro Candido Open the ORCID record for Leandro Candido [Opens in a new window] ,
Marek Cúth  and
Ondřej Smetana
Leandro Candido*
Affiliation:
Universidade Federal de S ao Paulo - UNIFESP, Instituto de Ciência e Tecnologia, Departamento de Matematica, São José dos Campos, SP, Brasil, (leandro.candido@unifesp.br)
Marek Cúth
Affiliation:
Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 83, 186 75 Prague 8, Czech Republic (cuth@karlin.mff.cuni.cz, ond.smetana@gmail.com)
Ondřej Smetana
Affiliation:
Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 83, 186 75 Prague 8, Czech Republic (cuth@karlin.mff.cuni.cz, ond.smetana@gmail.com)
*
*Corresponding author.
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Abstract

We exhibit a new approach to the proofs of the existence of a large family of almost isometric ideals in nonseparable Banach spaces and existence of a large family of almost isometric local retracts in metric spaces. Our approach also implies the existence of a large family of nontrivial projections on every dual of a nonseparable Banach space. We prove three possible formulations of our results are equivalent. Some applications are mentioned which witness the usefulness of our novel approach.

Keywords

Almost isometric idealsalmost isometric local retractsseparable reductionrich familymethod of suitable models

MSC classification

Primary: 46B07: Local theory of Banach spaces 46B26: Nonseparable Banach spaces
Secondary: 03C30: Other model constructions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 25
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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