Basis Sets in Banach Spaces

  • S. V. Konyagin
  • Y. V. Malykhin
Part of the Springer Optimization and Its Applications book series (SOIA, volume 68)


A set M in a linear normed space X over a field K (K=ℝ or K=ℂ) is called a basis set if every xX can be represented as a sum x=∑ k c k e k , where e k M, e k e l (kl), c k K∖{0}, ∑ k denotes either \(\sum_{k=1}^{\infty}\) or \(\sum_{k=1}^{N}\), and this representation is unique up to permutations. We prove the existence of an infinite-dimensional separable Banach space X with a basis set M such that no arrangement of M forms a Schauder basis.

Key words

Basis Schauder basis Rearrangement 

Mathematics Subject Classification

46Bxx 42A20 39B52 


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Function TheorySteklov Mathematical Institute, Russian Academy of SciencesMoscowRussia

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