= Degree ordering in an orthonormal polynomial basis
{title2=$\deg p_k=k$}
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 $\deg p_k=k$, or that the <polynomials> arise by applying the <Gram-Schmidt process> to $1,x,x^2,\ldots$ in that order. Under this <vertex degree>[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-<vertex degree>[degree] <polynomial>. The usual <Gaussian quadrature> construction therefore uses the <vertex degree>[degree]-ordered family; arbitrary relabelling is not enough.
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