Given a finite-dimensional inner product space V and a linear functional φ ε‎ V′, there exists w ε‎ V such that φ‎(v)=⟨v,w⟩ for all v ε‎ V. Further the norm ∥φ‎∥ of φ‎ equals the norm of w. The theorem extends to infinite-dimensional Hilbert spaces and their (analytic) dual spaces.