On approximate ℓ1 systems in Banach spaces

S. J. Dilworth, Denka Kutzarova, P. Wojtaszczyk

Research output: Contribution to journalArticlepeer-review


Let X be a real Banach space and let (f(n)) be a positive nondecreasing sequence. We consider systems of unit vectors (xi)i = 1 in X which satisfy ∥∑i∈ A ± xi∥ ≥ A - f(A), for all finite A⊂ ∌ and for all choices of signs. We identify the spaces which contain such systems for bounded (f(n)) and for all unbounded (f(n)). For arbitrary unbounded (f(n)), we give examples of systems for which [xi] is H.I., and we exhibit systems in all isomorphs of ℓ1 which are not equivalent to the unit vector basis of ℓ1. We also prove that certain lacunary Haar systems in L1 are quasi-greedy basic sequences.

Original languageEnglish (US)
Pages (from-to)214-241
Number of pages28
JournalJournal of Approximation Theory
Issue number2
StatePublished - 2002
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • General Mathematics
  • Applied Mathematics


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