Basis of a Hilbert space

Let \(\mch\) be a Hilbert space. An orthonormal sequence \((u_n)_{n\geq 1}\) in \(\mch\) is a basis of \(\mch\) if the linear span of \(\theset{u_n \suchthat n\geq 1}\), the set of finite linear combinations of the \(u_n\), is dense in \(\mch\).