There are far to be just curiosities.
Infinite dimensional Finsler manifolds are fundamental tools in Variational Calculus. There is a lot of old/recent papers dealing with the infinite dimensional Lusternik-Schnirelmann theory.
Just to select one (sorry for writing the hyperlink on three lines, but it seems it's the only way, I do not know why):
http://archive.numdam.org/ARCHIVE/AIHPC/AIHPC_1988__5_2/
AIHPC_1988__5_2_119_0/
AIHPC_1988__5_2_119_0.pdf
[and the references therein].
