Continuity of coordinate functionals of filter bases in Banach spaces

De Rancourt, Noé; Kania, Tomasz; Swaczyna, Jarosław

Document type :

Compte-rendu et recension critique d'ouvrage

DOI :

10.1016/j.jfa.2023.109869

Title :

Continuity of coordinate functionals of filter bases in Banach spaces

Author(s) :

De Rancourt, Noé [Auteur]
Univerzita Karlova [Praha, Česká republika] = Charles University [Prague, Czech Republic] [UK]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Kania, Tomasz [Auteur]
Institute of Mathematics of the Czech Academy of Science [IM / CAS]
Faculty of Mathematics and Computer Science of the Jagiellonian University
Swaczyna, Jarosław [Auteur]
Łódź University of Technology
Institute of Mathematics of the Czech Academy of Science [IM / CAS]

Journal title :

Journal of Functional Analysis

Pages :

109869

Publisher :

Elsevier

Publication date :

2023-05

ISSN :

0022-1236

English keyword(s) :

Filter bases in Banach spaces
Generalised bases
Automatic continuity
Definable filters

HAL domain(s) :

Mathématiques [math]/Analyse fonctionnelle [math.FA]

English abstract : [en]

We prove that the coordinate functionals associated with filter bases in Banach spaces are continuous as long as the underlying filter is analytic. This removes the large-cardinal hypothesis from the result established by ...
Show more >We prove that the coordinate functionals associated with filter bases in Banach spaces are continuous as long as the underlying filter is analytic. This removes the large-cardinal hypothesis from the result established by the two last-named authors ([Bull. Lond. Math. Soc. 53 (2021)]) at the expense of reducing the generality from projective to analytic. In particular, we obtain a ZFC solution to Kadets' problem of continuity of coordinate functionals associated with bases with respect to the filter of statistical convergence. Even though the automatic continuity of coordinate functionals beyond the projective class remains a mystery, we prove that a basis with respect to an arbitrary filter that has continuous coordinate functionals is also a basis with respect to a filter that is analytic.Show less >

Language :

Anglais

Popular science :

Non

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