Completeness of an orthonormal system ($\overline{\mathrm{span}\{e_n\}}=H$)

An orthonormal set is complete in a Hilbert space when the closure of its linear span is the whole space. Equivalently, the only vector orthogonal to every member is zero: project any vector onto the closed span, and its remaining orthogonal component must vanish. This is completeness of an expansion system, distinct from completeness of the ambient metric space.