Angles and Schauder basis in Hilbert spaces

TL;DR

The paper proves that Schauder bases in Hilbert spaces must have vectors with angles bounded away from zero and shows certain function systems cannot form Schauder bases in specific L2 spaces.

Contribution

It establishes a lower bound on angles between basis vectors in Hilbert spaces and rules out specific systems as Schauder bases in L2 spaces with discrete measures.

Findings

Angles between basis vectors have a positive lower bound.

Certain systems like rotated powers cannot be Schauder bases in L2 with discrete measures.

Provides conditions restricting Schauder basis constructions in Hilbert and L2 spaces.

Abstract

Let $\mathcal{H}$ be a complex separable Hilbert space. We prove that if $\{f_{n}\}_{n=1}^{\infty}$ is a Schauder basis of the Hilbert space $\mathcal{H}$, then the angles between any two vectors in this basis must have a positive lower bound. Furthermore, we investigate that $\{z^{\sigma^{-1}(n)}\}_{n=1}^{\infty}$ can never be a Schauder basis of $L^{2}(\mathbb{T},\nu)$, where $\mathbb{T}$ is the unit circle, $\nu$ is a finite positive discrete measure, and $\sigma: \mathbb{Z} \rightarrow \mathbb{N}$ is an arbitrary surjective and injective map.