Kernel-orthogonal proof of the Riesz representation theorem
= Kernel-orthogonal proof of the Riesz representation theorem
{c}
For a nonzero bounded functional $f$ on a Hilbert space, choose nonzero $z\in(\ker f)^\perp$. Then
$$
x-\frac{f(x)}{f(z)}z\in\ker f
$$
implies $f(x)=\langle x,f(z)z/\lVert z\rVert^2\rangle$ in the real case. Cauchy-Schwarz and evaluation on the representing vector give equality of the two norms.