Angles and Schauder basis in Hilbert 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.