Fischer inner product
= Fischer inner product
{c}
{title2=$\langle x^\alpha,x^\beta\rangle_F=\alpha!\,\delta_{\alpha\beta}$}
The Fischer inner product on <polynomials> is $\langle x^\alpha,x^\beta\rangle_F=\alpha!\,\delta_{\alpha\beta}$. Multiplication by $x_i$ is adjoint to differentiation by $x_i$. Therefore multiplication by $|x|^2$ is adjoint to the ordinary <Laplacian>, yielding the <harmonic decomposition of homogeneous polynomials> by finite-dimensional orthogonality.