Hilbert–Schmidt theorem

In mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations, it is very useful in solving elliptic boundary value problems.

Statement of the theorem
Let (H, ⟨, ⟩) be a real or complex Hilbert space and let A : H → H be a bounded, compact, self-adjoint operator. Then there is a sequence of non-zero real eigenvalues λi, i = 1, …, N, with N equal to the rank of A, such that |λi| is monotonically non-increasing and, if N = +∞, $$\lim_{i \to + \infty} \lambda_{i} = 0.$$

Furthermore, if each eigenvalue of A is repeated in the sequence according to its multiplicity, then there exists an orthonormal set φi, i = 1, …, N, of corresponding eigenfunctions, i.e., $$A \varphi_{i} = \lambda_{i} \varphi_{i} \mbox{ for } i = 1, \dots, N.$$

Moreover, the functions φi form an orthonormal basis for the range of A and A can be written as $$A u = \sum_{i = 1}^{N} \lambda_{i} \langle \varphi_{i}, u \rangle \varphi_{i} \mbox{ for all } u \in H.$$