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Spectral Theorem For Compact Self-Adjoint Operators . For t a compact, self adjoint operator on hilbert space h, t = p n λneλ n in which eλ n is the projection onto mn where mn is the eigenspace associated with λn. In this subsection, based on theorem 6.2, we are interested in approximating a compact operator by some operators with nite rank de ned below. NUS Math Module Review MA5206 Graduate Analysis II I Got Notes Lah! from igotnoteslah.com This is about the spectral decomposition of compact operators. Let tbe a continuous linear map v !v for a (separable) hilbert space v. T2l(e;f) is an operator of nite rank if r(t) is nite dimensional.