In a finite dimensional space, one calculates the Lie derivative as
$L_f(g)(x) = \langle \nabla g, f \rangle$
What is the equivalent in an infinite dimensional space? For example if $g$ takes as argument a function and $f$ is an infinite dimensional vector, how does one think about the Lie derivative?
I am familiar with the Gateau derivative, so does one simply replace the gradient with this? Then we might have
$L_f(g)(x) = \langle dg, f \rangle$
for some appropriate inner product? for instance $L^2$?