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Which is the space of mappings from L^2 to itself?

  1. Feb 2, 2015 #1
    Hi there,

    I was wondering, which is the space of (not necessarily linear) mappings from ##L^2## to itself? If you have an element ##f(x) \in L^2##, then a nonlinear mapping could be ##g(\cdot)##. Then if ##g## is bounded the image is in ##L^2##, does that mean that the space of linear and nonlinear mappings is also ##L^2##? Could you consider that an endomorphism if it is not linear? Is that space smaller than the space of bounded operators from ##L^2## to itself?

    Thank you
     
  2. jcsd
  3. Feb 2, 2015 #2

    mathman

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    In order for a space to be [itex]L^2[/itex] its domain must be a measure space so that integrals can be defined. This is undefined for the domain of these mappings.
     
  4. Feb 2, 2015 #3
    You are right. Thank you
     
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