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arXiv · 1806.07866

Angles and Schauder basis in Hilbert spaces

Bingzhe Hou·Yang Cao·Geng Tian·Xinzhi Zhang
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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.

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Bingzhe Hou, Yang Cao, Geng Tian, Xinzhi Zhang. 2018-06-20. Angles and Schauder basis in Hilbert spaces. https://arxiv.org/abs/1806.07866

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