Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 11H / a

Bilinearity and symmetry are immediate. Since $r\ge 0$, $⟨u,u⟩\ge 0$. If this is zero, continuity of $u^{2}r$ makes it identically zero. The points where $r>0$ are dense because only finitely many are excluded, so $u$ vanishes on a dense set and hence everywhere. Thus the form is an inner product.