 When "H" is separable, one can mix the basis {"en"} with a countable orthonormal basis for the kernel of "T", and obtain an orthonormal basis {"fn"} for "H", consisting of eigenvectors of "T" with real eigenvalues {"μn"} such that .Corollary For every compact self-adjoint operator "T" on a real or complex separable infinite-dimensional Hilbert space "H", there exists a countably infinite orthonormal basis {"fn"} of "H" consisting of eigenvectors of "T", with corresponding eigenvalues , such that .Proving the spectral theorem for a Hermitian "n" × "n" matrix "T" hinges on showing the existence of one eigenvector "x".