Angles and Schauder basis in Hilbert spaces

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 $L^{2}(mathbb{T}, u)$, where $mathbb{T}$ is the unit circle, $ u$ is a finite positive discrete measure, and $sigma: mathbb{Z} rightarrow mathbb{N}$ is an arbitrary surjective and injective map.