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" = +∞,
formula_1
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.,
formula_2
Moreover, the functions "φ""i" form an orthonormal basis for the range of "A" and "A" can be written as
formula_3