Solution
= Solution
The <Riesz representation theorem> says every bounded linear functional $f$ on a Hilbert space has a unique $z$ with $f(x)=\langle x,z\rangle$ and $\|f\|=\|z\|$. For $f\ne0$, decompose relative to $\ker f$; its one-dimensional orthogonal complement supplies $z$, and Cauchy-Schwarz proves the norm statement and uniqueness.