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Math Help - spectral theorem

  1. #1
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    spectral theorem

    Suppose V is a finite-dimensional inner product space over C and T:V->V is a self-adjoint linear transformation. Then V has an orthonormal basis consisting of eigenvectors of T.
    What would happens if V is infinite-dimensional?
    Does the proof of finite-dimensional still work for infinite-dimensional? Why?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by guin View Post
    Suppose V is a finite-dimensional inner product space over C and T:V->V is a self-adjoint linear transformation. Then V has an orthonormal basis consisting of eigenvectors of T.
    What would happens if V is infinite-dimensional?
    Does the proof of finite-dimensional still work for infinite-dimensional? Why?
    It almost works, but you also need T to be compact. See wikipedia, and the book `functional analysis' by Rudin should cover this (although I don't have a copy to hand).

    Curiously, this question is actually a question about functional analysis; you've left the field of algebra with one small step! I would suggest asking any further questions in the analysis forum...
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