Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Almost Auerbach, Markushevich and Schauder bases in Hilbert and Banach spaces

Anton Tselishchev

Abstract

For any sequence of positive numbers $(\varepsilon_n)_{n=1}^\infty$ such that $\sum_{n=1}^\infty \varepsilon_n = \infty$ we provide an explicit simple construction of $(1+\varepsilon_n)$-bounded Markushevich basis in a separable Hilbert space which is not strong, or, in other terminology, is not hereditary complete; this condition on the sequence $(\varepsilon_n)_{n=1}^\infty$ is sharp. Using a finite-dimensional version of this construction, Dvoretzky's theorem and a construction of Vershynin, we conclude that in any Banach space for any sequence of positive numbers $(\varepsilon_n)_{n=1}^\infty$ such that $\sum_{n=1}^\infty \varepsilon_n^2 = \infty$ there exists a $(1+\varepsilon_n)$-bounded Markushevich basis which is not a Schauder basis after any permutation of its elements.

Almost Auerbach, Markushevich and Schauder bases in Hilbert and Banach spaces

Abstract

For any sequence of positive numbers such that we provide an explicit simple construction of -bounded Markushevich basis in a separable Hilbert space which is not strong, or, in other terminology, is not hereditary complete; this condition on the sequence is sharp. Using a finite-dimensional version of this construction, Dvoretzky's theorem and a construction of Vershynin, we conclude that in any Banach space for any sequence of positive numbers such that there exists a -bounded Markushevich basis which is not a Schauder basis after any permutation of its elements.

Paper Structure

This paper contains 9 sections, 4 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. The main definitions
  3. Formulations of the results
  4. Proof of Theorem 1
  5. Completeness
  6. Biorthogonality
  7. The basis is not strong
  8. The norm estimate
  9. Proof of Theorem 2

Key Result

Theorem \oldthetheorem

For any sequence of nonnegative numbers $(\varepsilon_n)_{n\ge 1}$ such that $\sum_{n=1}^\infty \varepsilon_n = \infty$ there exists a $(1+\varepsilon_n)$-bounded M-basis in $\ell^2$ which is not strong.

Theorems & Definitions (4)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary 1
  • Corollary 2