Compact elliptic spectral theorem
= Compact elliptic spectral theorem
On a <closed manifold> with a <Riemannian metric> the <positive Laplace-Beltrami operator> is <self-adjoint> with <compact resolvent>. It has a complete smooth <orthonormal eigenbasis>, finite-dimensional <eigenspaces>, nonnegative <eigenvalues> tending to infinity, and polynomial eigenvalue-counting bounds. <Elliptic regularity> bounds derivatives of its <eigenfunctions> polynomially in their <eigenvalues>.