Reactive programming

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

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

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 (Theorem 8.94)