Finite-evaluation inner product on polynomials ($⟨p,q⟩={\sum }_{j=0}^n{w}_{j}p(x_j)q(x_j)$)

At distinct real nodes $x_0,\dots ,x_n$ with positive weights $w_j$, the displayed expression is an inner product on the real vector space of polynomials of degree at most $n$. Symmetry and bilinearity follow from finite sums. If its squared norm vanishes, positivity of every weight forces $p(x_j)=0$ at every node. The root bound for a nonzero polynomial then forces $p=0$, proving positive definiteness.

More generally, on any vector subspace of real functions, the evaluation form is an inner product exactly when the evaluation map $p\mapsto (\sqrt{w_j}p(x_j){)}_j$ is injective. On polynomials of degree at most $n+1$ it fails: the nonzero node polynomial ${\prod }_j(x-x_j)$ vanishes at all nodes. This gives a positive semidefinite form with a nontrivial kernel. On the smaller polynomial space, the Lagrange interpolation polynomial identifies evaluation coordinates with ${\mathbb{R}}^{n+1}$, and Cauchy-Schwarz inequality follows either from the Euclidean inequality in those coordinates or from minimizing $\Vert p-tq{\Vert }^2$ over real $t$.