Product Laplacian eigenspace decomposition ($E_{\lambda }(M\times N)={⨁}_{\mu +\nu =\lambda }{E}_{\mu }(M)\otimes E_{\nu }(N)$)

For closed manifolds with the product Riemannian metric, the positive Laplace-Beltrami operator is the sum of the two factor operators. Products of their orthonormal eigenbases form a complete basis, and eigenvalues add. Hence the product eigenspace at $\lambda$ is the displayed finite orthogonal sum of tensor products. The discrete assertion requires a discrete spectral setting.