= Discrete sampling inner product on polynomial spaces
{title2=$\langle f,g\rangle=\sum_{j=0}^{n}w_jf(t_j)g(t_j)$}
Let $P_n(\mathbb R)$ be the <vector space> of real <polynomials> of <degree of a polynomial> at most $n$. Choose $n+1$ distinct real sample points $t_0,\ldots,t_n$ and positive real weights $w_0,\ldots,w_n$. Then
$$
\langle f,g\rangle=\sum_{j=0}^{n}w_jf(t_j)g(t_j)
$$
is an <inner product>. Symmetry and <bilinearity> follow from <linearity> of <polynomial evaluation>. Its value at $(f,f)$ is nonnegative, and equality forces all $f(t_j)=0$. By the <root bound for a polynomial>, a nonzero <polynomial> of <degree of a polynomial> at most $n$ cannot vanish at $n+1$ distinct points. Thus the form has <positive definiteness>.
Equivalently, the <polynomial evaluation map on a finite point set> identifies $P_n(\mathbb R)$ with $\mathbb R^{n+1}$, equipped with a <weighted Euclidean inner product>. Too few sample points destroy <positive definiteness>: if the distinct points number at most $n$, the nonzero <polynomial> $\prod_j(x-t_j)$ lies in $P_n(\mathbb R)$ and has zero sampled squared <norm>.
For three equally weighted samples at $-1,0,1$, an <orthonormal basis> is
$$
1,\qquad\sqrt{\frac32}\,x,\qquad\frac{3x^2-2}{\sqrt2}.
$$
This construction gives a finite sampling version of <orthogonal polynomials> and permits exact <Gram-Schmidt orthogonalization> using only finitely many evaluations.
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