# 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 $L^2$?

You might find what you're looking for in one of the following links:

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

Basically, consider looking for material on infinite dimensional manifolds.

• Well, thanks, but, I would like some attempt at an answer here. Aug 24 '17 at 21:40
• Why not just try reading the sections covering the Lie derivative in any of those books?
– g.s
Aug 26 '17 at 5:47