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

## 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

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.