I understand the definition of the Fréchet derivative. However, outside of functions on $\mathbb R^n$, I've never encountered an application where it was particularly useful.
Can anyone share examples of (or references to) uses of the Fréchet derivative, especially ones where other methods would not have worked?
Optimisation problems are preferred, but nice applications elsewhere would also be appreciated. As suggested above, I am looking for examples where the domain is not $\mathbb R^n$.
Some Background
For example, on the odd occasion where I've had to find the optimizer of some functional defined on a Banach space, I always try to calculate the Fréchet derivative to see if it would help. I usually end up with a condition on variations that I don't really know how to make use of, and end up resorting to other methods to solve my problem.
It doesn't help that the references I turn to almost exclusively only have examples of the use of the Fréchet derivative on finite-dimensional spaces. If not, then they were actually optimising over a space of appropriately smooth functions, so Euler-Lagrange would have done the trick.
I suppose this is just a lack of education on my part, but I do think it's noteworthy that the Wikipedia article I link to above only features examples in the form of calculating the derivative of specific functions. It says very little about why I might want to calculate it, aside from passing references in the introduction, or, how, once I've calculated it, I might be able to make use of it.
The books Introduction to Mechanics and Symmetry and Manifolds, Tensor Analysis and Applications discuss some applications to physics of the Fréchet derivative outsise $\mathbb{R}^{n}$. The latter uses Fréchet derivatives in the context of Classical Field theory and variational problems. The former uses some of these concepts in classical mechanics.