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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