Let \(H\) be an inner product space with inner product \(\inner{\cdot}{\cdot}\) and norm \(\norm{u}\coloneqq\sqrt{\inner{u}{u}}\), and let \(I\) be a countable index set. A family \((u_i)_{i\in I}\) of elements of \(H\) is orthonormal if
\(\inner{u_i}{u_j} = 0\) for all \(i,j\in I\) with \(i \neq j\), and
\(\norm{u_j} = 1\) for all \(j\in I\).