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

Abstract

For any sequence of positive numbers ${\left({\epsilon }_n\right)}_{n=1}^{\infty }$ such that ${\sum }_{n=1}^{\infty }{\epsilon }_n=\infty$ we provide an explicit simple construction of $(1+{\epsilon }_n)$-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 ${\left({\epsilon }_n\right)}_{n=1}^{\infty }$ 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 ${\left({\epsilon }_n\right)}_{n=1}^{\infty }$ such that ${\sum }_{n=1}^{\infty }{\epsilon }_n^2=\infty$ there exists a $(1+{\epsilon }_n)$-bounded Markushevich basis which is not a Schauder basis after any permutation of its elements.