It's not too bad to formalize :) The [https://en.m.wikipedia.org/wiki/Fixed-point_iteration](wikipedia https://en.m.wikipedia.org/wiki/Fixed-point_iteration](wi

It's not too bad to formalize :) The [https://en.m.wikipedia.org/wiki/Fixed-point_iteration](wikipedia https://en.m.wikipedia.org/wiki/Fixed-point_iteration](wikip... article on fixed point iteration) gives lip service to everything working fine in arbitrary metric spaces. It's easy to define a useful metric between functions (e.g. via integrating their difference, ignoring edge cases like occasional pointwise differences since a more careful treatment can give a presentation where those don't matter for the problem at hand), so the same kinds of theorems that work with cos(x) on the reals also work with differential operators on function spaces.