Riesz representation theorem for Hilbert spaces

Let \(H\) be a Hilbert space over \(\CC\) with inner product \(\inner{\cdot}{\cdot}\) linear in the first argument, and let \(\varphi \in \dualof{H}\) be an element of its dual. Then there exists a unique \(f\in H\) such that \(\varphi(x) = \inner{x}{f}\) for all \(x\in H\), and \(\norm{f}_H = \norm{\varphi}_{\dualof{H}}\), the dual norm of \(\varphi\).