Angles and Schauder basis in Hilbert spaces

Let $\mathcal{H}$ be a complex separable Hilbert space. We prove that if $\{f_n{\}}_{n=1}^{\mathrm{\infty }}$ 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 $\{z^{{\sigma }^{-1}(n)}{\}}_{n=1}^{\mathrm{\infty }}$ 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.