Let $H$ be a non-separable Hilbert space. Let $\{ \phi _\alpha\} _{\alpha \in A} $ be a orthonormal system on $H$.
Show that for every $x\in H$, there are only countably many Fourier coefficients, i.e. there are countably many distinct $(x,\phi _\alpha)$ only
I have no idea how to solve this; a non-separable space DOES NOT have a countable dense subset, I don't see how this definition can be applied to deduce there ARE countably many Fourier coefficients.
