What is Hilbert basis?

A family $(e_i)_{i\in I}$ is orthonormal if $\langle e_i, e_j\rangle = \delta_{ij}$, and a Hilbert basis (orthonormal basis) if moreover its finite linear combinations are dense in $H$ (the family is total). We treat the countable case $I = \N$, which by Gram–Schmidt covers every separable $H$ (Proposition 13.8).