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Interchanging Linear Operator and Infinite Sum

  1. Oct 12, 2013 #1
    Suppose that [itex]x\in H[/itex], where H is a Hilbert space. Then x has an orthogonal decomposition [itex]x = \sum_{i=0}^\infty x_i[/itex].

    I have a linear operator P (more specifically a projection operator), and I want to write:
    [itex]P(x) = \sum_{i=0}^\infty P(x_i)[/itex].

    How can I justify taking the operator inside the infinite sum?
  2. jcsd
  3. Oct 12, 2013 #2


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    Science Advisor

    This is true if the linear operator P is bounded. Otherwise, it might be false.
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