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

  • #1
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Main Question or Discussion Point

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
 

Answers and Replies

  • #2
mathman
Science Advisor
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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.
 
  • #3
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You are right. Thank you
 

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