On Schauder Bases in Hilbert Space

TL;DR

This paper generalizes a classical result in Hilbert spaces, showing that two orthonormal sequences that are quadratically close share the basis property, extending the understanding of basis stability under perturbations.

Contribution

It introduces a broad generalization of the known criterion for basis equivalence of orthonormal sequences in Hilbert spaces.

Findings

If one orthonormal sequence is a basis, the quadratically close sequence is also a basis.

The result applies to a wider class of sequences beyond the classical case.

Provides a new perspective on stability of bases under perturbations.

Abstract

In this short note we present a far generalization of the following very well-known assertion: assume that we have two orthonormal sequences in a Hilbert space and these sequences are quadratically close to each other. Then if one of these sequences is a basis in the Hilbert space then so is the other one.