Angles and Schauder basis in Hilbert spaces

• DOI: 10.48550/arxiv.1806.07866
• Publication Date: 2018-01-01
• Authors: Bingzhe Hou; Yang Cao; Geng Tian; Xinzhi Zhang
• Keywords: Basis (linear algebra); Schauder basis; Surjective function; Unit circle

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