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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 A u = \sum_{i = 1}^{N} \lambda_{i} \langle \varphi_{i}, u \rangle \varphi_{i} \mbox{ for all } u \in H.
References
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