 Solution

The Riesz representation theorem says every bounded linear functional $f$ on a Hilbert space has a unique $z$ with $f(x)=⟨x,z⟩$ and $\Vert f\Vert =\Vert z\Vert$. For $f\ne 0$, decompose relative to $\ker f$; its one-dimensional orthogonal complement supplies $z$, and Cauchy-Schwarz proves the norm statement and uniqueness.