Spectral Theorem for Compact Selfadjoint Operators

Let $A$ be a compact self-adjoint operator on a separable Hilbert space $H$. Then $H$ has an orthonormal basis of eigenvectors of $A$. Every eigenvalue is real, each nonzero eigenvalue has finite multiplicity, and the nonzero eigenvalues form a finite sequence or a sequence tending to $0$. The kernel of $A$ is the zero-eigenspace and may be infinite-dimensional.