orthonormal basis of infinite dimensional Hilbert space H is not a basis of H as vector space? Apparently the orthonormal basis $(e_n)_{n\in \mathbb{N}}$ of the Hilbert space $H$ (in special case, infinitly dimensional) is not a basis of $H$ as a vectorspace. Is there a way to prove this?
Every vector space basis of an infinite dimensional Banach or Hilbert space is necessarily uncountable. This is actually a non-trivial result and follows from the Baire category theorem, see e.g. Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.
A basis means that for everu $u\in H$ you have $u=a_{i_1}e_{i_1}+...+a_{i_n}e_{i_n}$ for $n$ finite, and orthogonal basis means that $u=\sum_{n\geq 0}a_ne_n$ where the sequence $\sum_{i=0}^{i=n}a_ie_i$ converges, so for orthogonal basis, all the coefficients may not be zero.