Orthonormal system of simultaneous eigenvectors

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Suppose we have a commutative family of compact, self-adjoint operators on a Hilbert space. Prove that there is an orthonormal system of simultaneous eigenvectors for the family.

I'm not sure how to approach this problem. Any hints would be appreciated.

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1 Answer

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Since the family is commutative, the operators are simultaneously diagonalizable if one operator is diagonalizable, so it suffices to find a single set of orthonormal eigenvectors for one self-adjoint operator of the family. By spectral theorem, any compact self-adjoint operator on real/complex hilbert space is diagonalizable, so we proved what we want.

Check this page about spectral theorem

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