Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor

    This is true if the linear operator P is bounded. Otherwise, it might be false.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook