Counterexamples in isometric theory of symmetric and greedy bases

Fernando Albiac, José L. Ansorena, Óscar Blasco, Hùng Việt Chu, Timur Oikhberg

Research output: Contribution to journalArticlepeer-review


We continue the study initiated in Albiac and Wojtaszczyk (2006) of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to 1. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A) but fails to be 1-suppression unconditional, thus settling Problem 4.4 from Albiac and Ansorena (2017). In particular, our construction demonstrates that bases with Property (A) need not be 1-greedy even with the additional assumption that they are unconditional and symmetric. We also exhibit a finite-dimensional counterpart of this example, and show that, at least in the finite-dimensional setting, Property (A) does not pass to the dual. As a by-product of our arguments, we prove that a symmetric basis is unconditional if and only if it is total, thus generalizing the well-known result that symmetric Schauder bases are unconditional.

Original languageEnglish (US)
Article number105970
JournalJournal of Approximation Theory
StatePublished - Jan 2024


  • Greedy basis
  • Property (A)
  • Suppression unconditional basis
  • Symmetric basis
  • Thresholding greedy algorithm

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • General Mathematics
  • Applied Mathematics


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