Typically the space $L[0,\pi]$ is said to have a orthogonal basis that consists of the harmonics of sine and cosine, i.e. $\{1,\sin(n\omega t),\cos(n\omega t)\ |\ n\in\mathbb{N}\}$.
However you can also think of the Fourier spectrum of a signal in terms of frequency and phase rather than sines and cosines. Does that mean that there is an uncountable (non-orthogonal) basis that consists of $\{1,\sin(n\omega t-\phi)\ |\ n\in\mathbb{N}, \phi\in[0,\pi]\}$?
Does it even make sense to think about uncountable bases? Or does it open some kinda can of worms?