# What is the infinite dimensional counterpart of the Lie derivative?

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 $L^2$?