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

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

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.