Let (H,⟨⋅,⋅⟩)\left( H, \left\langle \cdot,\cdot \right\rangle \right)(H,⟨⋅,⋅⟩) be a Hilbert space. For linear functionals f∈H∗f \in H^{ \ast }f∈H∗ and x∈H\mathbf{x} \in Hx∈H satisfying f(x)=⟨x,w⟩f ( \mathbf{x} ) = \left\langle \mathbf{x} , \mathbf{w} \right\ranglef(x)=⟨x,w⟩ and ∥f∥H∗=∥w∥H\| f \|_{H^{\ast}} = \| \mathbf{w} \|_{H}∥f∥H∗=∥w∥H, there exists a unique w∈H\mathbf{w} \in Hw∈H.
