TITLE: What is the infinite dimensional counterpart of the Lie derivative?
QUESTION [6 upvotes]: 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 $L^2$?

REPLY [4 votes]: You might find what you're looking for in one of the following links:

Lang:  http://www.springer.com/gp/book/9780387985930
Kriegl & Michor:  http://bookstore.ams.org/surv-53
Choquet-Bruhat & DeWitt-Morette: 
https://www.elsevier.com/books/analysis-manifolds-and-physics-revised-edition/choquet-bruhat/978-0-444-86017-0
Omori:  http://bookstore.ams.org/mmono-158/

Basically, consider looking for material on infinite dimensional manifolds.