Building highly conditional almost greedy and quasi-greedy bases in Banach spaces

F. Albiac, J. L. Ansorena, S. J. Dilworth, Denka Kutzarova

Research output: Contribution to journalArticlepeer-review


It is known that for a conditional quasi-greedy basis B in a Banach space X, the associated sequence (km[B])m=1 of its conditionality constants verifies the estimate km[B]=O(log⁡m) and that if the reverse inequality log⁡m=O(km[B]) holds then X is non-superreflexive. Indeed, it is known that a quasi-greedy basis in a superreflexive quasi-Banach space fulfils the estimate km[B]=O(log⁡m)1−ϵ for some ϵ>0. However, in the existing literature one finds very few instances of spaces possessing quasi-greedy basis with conditionality constants “as large as possible.” Our goal in this article is to fill this gap. To that end we enhance and exploit a technique developed by Dilworth et al. in [15] and craft a wealth of new examples of both non-superreflexive classical Banach spaces having quasi-greedy bases B with km[B]=O(log⁡m) and superreflexive classical Banach spaces having for every ϵ>0 quasi-greedy bases B with km[B]=O(log⁡m)1−ϵ. Moreover, in most cases those bases will be almost greedy.

Original languageEnglish (US)
Pages (from-to)1893-1924
Number of pages32
JournalJournal of Functional Analysis
Issue number6
StatePublished - Mar 15 2019


  • Almost greedy basis
  • Conditionality constants
  • Quasi-greedy basis
  • Subsymmetric basis

ASJC Scopus subject areas

  • Analysis


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