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Mean ergodic theorem von Neumann

  1. Jan 21, 2005 #1
    I wonder If someone could state the mean ergodic theorem von neumann with out using meassure spaces ? I have studied normed spaces, banach spaces and hilbert spaces, that is complete normed inner product spaces.

    Could someone state and explain the theorem for me? :smile:
  2. jcsd
  3. Jan 21, 2005 #2

    matt grime

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    As ergodic theory is the study of invariants measurable spaces, in some loose sense, this doesn't seem very likely. However, reading a statement of it I don't understand what the measure theoretic objection is. One can translate to little l 2 and it states that, modulo some dodgy rearranging of terms, we have approximately:

    given an isometry of hilbert space l_2, T, and an element a in l_2, there is a b in l_2 such that

    [tex] \frac{1}{n} \sum_{k=0}^n <T^ka,b>^2[/tex]

    tends to zero as n tends to infinity.
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