Orthonormal polynomial ($({\varphi }_r,{\varphi }_s)={\delta }_{rs}$)

An orthonormal family of orthogonal polynomials also gives each member unit norm in the chosen weighted inner product. Its pairwise inner products are the Kronecker delta. Dividing every nonzero orthogonal polynomial by its norm produces such a family; the sign or complex phase still requires a convention, such as a positive real leading coefficient.

An orthonormal basis of the polynomial space ${\Pi }_n$ need not be indexed by polynomial degree. Permuting the normalized Legendre polynomials, for example, preserves orthonormality but changes the degrees attached to the labels. Results about the number of polynomial roots of $p_k$ require the extra condition $\mathrm{deg}{p}_k=k$, or that the polynomials arise by applying the Gram-Schmidt process to $1,x,x^2,\dots$ in that order. Under this degree ordering, $p_k$ is orthogonal to every polynomial of lower polynomial degree because those earlier basis members span ${\Pi }_{k-1}$.

For a positive integration weight on a nondegenerate interval, this implies that $p_k$ has $k$ distinct polynomial roots in the open interval. If fewer than $k$ roots change sign there, multiply $p_k$ by the product of their linear factors. The product has constant nonzero sign except at its finitely many zeros, so its weighted integral cannot vanish, contradicting orthogonality to this lower-degree polynomial. The usual Gaussian quadrature construction therefore uses the degree-ordered family; arbitrary relabelling is not enough.