I’m trying to figure out what the gradient operator on L^2 is from the definition that for any vector field X, the gradient operator is such that
$\langle \nabla f, X \rangle = \partial_X f.$
Now, what I’m trying to derive in particular is that if $f(x)=\int U(x)$ then the L^2 gradient of the $f(x)$ is just $U’(x)$. I am mainly stuck on applying this definition, in particular on what I should pick as a vector field.