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Basis Sets in Banach Spaces

In: Nonlinear Analysis

Author

Listed:
  • S. V. Konyagin

    (Steklov Mathematical Institute, Russian Academy of Sciences)

  • Y. V. Malykhin

    (Steklov Mathematical Institute, Russian Academy of Sciences)

Abstract

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 x∈X can be represented as a sum x=∑ k c k e k , where e k ∈M, e k ≠e l (k≠l), 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.

Suggested Citation

  • S. V. Konyagin & Y. V. Malykhin, 2012. "Basis Sets in Banach Spaces," Springer Optimization and Its Applications, in: Panos M. Pardalos & Pando G. Georgiev & Hari M. Srivastava (ed.), Nonlinear Analysis, edition 127, chapter 0, pages 381-386, Springer.
    • Handle: RePEc:spr:spochp:978-1-4614-3498-6_23
    DOI: 10.1007/978-1-4614-3498-6_23
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