Hilbert–Schmidt theorem
# 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 $λ_i,$ i = 1, …, N, with N equal to the rank of A, such that $|λ_i|$ 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 $φ_i,$ i = 1, …, N, of corresponding eigenfunctions, i.e.,

$A\varphi _{i}=\lambda _{i}\varphi _{i}{\mbox{ for }}i=1,\dots ,N.$

Moreover, the functions $φ_i$ form an orthonormal basis for the range of A and A can be written as

$Au=\sum _{i=1}^{N}\lambda _{i}\langle \varphi _{i},u\rangle \varphi _{i}{\mbox{ for all }}u\in H.$

