[Math] Hamel basis and orthonormal basis for Hilbert spaces

Question

My question:

why don't we get a Hamel basis (a maximal linearly independent set) instead of a maximal orthonormal set for a Hilbert space. In what dimension can we use a Hamel basis and in which we can't?

Answer 1

We know the existence of a dense countable orthonormal set i.e. $\fbox{1}\langle x,e_i\rangle \,$ for $\forall i \in \mathbb{N}$ $\, \Rightarrow x=0.\,$
Since a Hamel basis on a Hilbert space is uncountable we can extend the linearly independent set $ \{ e_i\ | \forall i \in \mathbb{N} \}\,$ to a Hamel basis in a non trivial way. So $\fbox{1} \,$ yields a contradiction.