= Monomial orthonormal basis on the unit torus
{title2=$\langle z^\alpha,z^\beta\rangle=\delta_{\alpha\beta}$}
Integrate a <polynomial> times the conjugate of another over the product of unit circles, with normalized product angular measure. <Fourier orthogonality> makes the <monomials> $z_1^{\alpha_1}\cdots z_d^{\alpha_d}$ an <orthonormal basis> of the polynomial <vector space>. The <Hermitian form> is positive definite because the squared norm is the sum of squared absolute values of the finitely many coefficients. No completion or infinite series is needed to obtain this algebraic basis.
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