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Barrelled Countable Enlargements and the Bounding Cardinal

1996; Wiley; Volume: 53; Issue: 1 Linguagem: Inglês

10.1112/jlms/53.1.158

1469-7750

Stephen A. Saxon, Luis Manuel Sánchez Ruiz,

Advanced Banach Space Theory

Journal of the London Mathematical SocietyVolume 53, Issue 1 p. 158-166 Notes and papers Barrelled Countable Enlargements and the Bounding Cardinal S. A. Saxon, S. A. Saxon Department of Mathematics, University of Florida, PO Box 8000, Gainesville, Florida 32611-8000, USASearch for more papers by this authorL. M. Sánchez Ruiz, L. M. Sánchez Ruiz EUITI-Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, 46071 Valencia, Spain This paper was started while the second author stayed at the University of Florida supported by DGICYT BE91-332, later being partially supported by DGICYT PB91-0407.Search for more papers by this author S. A. Saxon, S. A. Saxon Department of Mathematics, University of Florida, PO Box 8000, Gainesville, Florida 32611-8000, USASearch for more papers by this authorL. M. Sánchez Ruiz, L. M. Sánchez Ruiz EUITI-Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, 46071 Valencia, Spain This paper was started while the second author stayed at the University of Florida supported by DGICYT BE91-332, later being partially supported by DGICYT PB91-0407.Search for more papers by this author First published: February 1996 https://doi.org/10.1112/jlms/53.1.158Citations: 5 AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract Replacing c with the bounding cardinal 𝔟 improves two standard BCE results of Robertson, Tweddle and Yeomans, and 𝔟 is optimal for the BCE codimension/inheritance result of Bonet and Pérez Carreras. Indeed, 𝔟 is the smallest infinite-dimensionality for metrizable barrelled spaces, and is the largest cardinal κ such that every subspace of codimension less than κ in a metrizable barrelled space is itself barrelled. We thus reconfirm 𝔟 as one of two 'optimal cardinals for metrizable barrelled spaces'. Esthetically pleasing, these properties of 𝔟 immediately solve the normable BCE problem without extra-ZFC axiomatic assumptions, reduce the separable quotient problem to Banach spaces E with 𝔟 ⩽ density character of E ⩽ c, and are part of the solution to one version of the metrizable BCE problem found in our sequel, devoted to the other 'optimal cardinal'. Citing Literature Volume53, Issue1February 1996Pages 158-166 RelatedInformation