Sear 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 = +∞, 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., Moreover, the functions φi form an orthonormal basis for the range of A and A can be written as References Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations (htt ps://archive.org/details/introductiontopa00roge) . Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. 356 (https://archive.org/details/introductiontopa 00roge/page/n370) . ISBN 0-387-00444-0. (Theorem 8.94) Retrieved from "https://en.wikipedia.org/w/index.php? title=Hilbert– Schmidt_theorem&oldid=1029304525" Last edited 10 months ago by Hellacioussatyr
