Finite polynomial-chaos orthogonal projection

For real $u\in L^2(\rho )$ and a finite subset $\mathrm{\Lambda }$ of the admissible degree box, with $({\psi }_{\alpha }{)}_{\alpha \in \mathrm{\Lambda }}$ as in Finite orthonormal polynomial tensor basis, define $c_{\alpha }=⟨u,{\psi }_{\alpha }{⟩}_{L^2(\rho )}=\int u(y){\psi }_{\alpha }(y)d\rho (y),P_{\mathrm{\Lambda }}u={\sum }_{\alpha \in \mathrm{\Lambda }}{c}_{\alpha }{\psi }_{\alpha }\mathrm{.}$ The scalar integral pairing is choice-free: for representatives $f,g$ of classes in $L^2(\rho )$, Cauchy–Schwarz makes $fg$ integrable, and the integral is unchanged when either representative is changed on a null set. Its bilinearity follows from linearity of the integral and its symmetry from commutativity of multiplication; moreover $\int f^{2}d\rho =\mathrm{\parallel }[f]{\mathrm{\parallel }}_2^2$, so it is positive definite on the quotient. These facts use only the integral pairing, not completeness of $L^2(\rho )$. The displayed finite sum is the orthogonal projection onto ${\mathcal{P}}_{\mathrm{\Lambda }}$ of Finite orthonormal polynomial tensor basis.