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.

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

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