Open AccessA dichotomy for subsymmetric basic sequences with applications to Garling spaces
Author(s) -
Fernando Albiac,
José L. Ansorena,
S. J. Dilworth,
Denka Kutzarova
Publication year - 2021
Publication title -
transactions of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.798
H-Index - 100
eISSN - 1088-6850
pISSN - 0002-9947
DOI - 10.1090/tran/8278
Subject(s) - mathematics , banach space , sequence (biology) , pure mathematics , interpolation space , fréchet space , functional analysis , basis (linear algebra) , eberlein–šmulian theorem , discrete mathematics , lp space , biochemistry , chemistry , genetics , gene , biology , geometry
Our aim in this article is to contribute to the study of the structure of subsymmetric basic sequences in Banach spaces (even, more generally, in quasi-Banach spaces). For that we introduce the notion of positionings and develop new tools which lead to a dichotomy theorem that holds for general spaces with subsymmetric bases. As an illustration of how to use this dichotomy theorem we obtain the classification of all subsymmetric sequences in certain types of spaces. To be more specific, we show that Garling sequence spaces have a unique symmetric basic sequence but no symmetric basis and that these spaces have a continuum of subsymmetric basic sequences.