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

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SUMMARY

The discussion centers on the space of mappings from L² to itself, specifically addressing both linear and nonlinear mappings. It establishes that for a mapping g(·) to be considered within the L² space, it must be bounded, ensuring the image remains in L². The conversation also touches on the concept of endomorphisms, questioning whether nonlinear mappings can qualify as such. Ultimately, it concludes that the space of these mappings is not necessarily equivalent to the space of bounded operators from L² to itself due to the requirement of a defined measure space for integrals.

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jorgdv
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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
 
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In order for a space to be L^2 its domain must be a measure space so that integrals can be defined. This is undefined for the domain of these mappings.
 
You are right. Thank you
 

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