 Kernel-orthogonal proof of the Riesz representation theorem

For a nonzero bounded functional $f$ on a Hilbert space, choose nonzero $z\in (\ker f{)}^{\perp }$. Thenimplies $f(x)=⟨x,f(z)z/\Vert z{\Vert }^2⟩$ in the real case. Cauchy-Schwarz and evaluation on the representing vector give equality of the two norms.