In the 50's through 70's there was a lot of research into the group of orientation-preserving homeomorphisms of the plane, denoted as Homeo$^+(\mathbb{R}^2)$ (in the compact-open topology, which in this case is just the topology of compact convergence). Long proofs were used to prove certain local properties like local arc-connectedness and local contractibility.
But much more recently it was shown that Homeo$^+(\mathbb{R}^2)$ is actually a Hilbert manifold, i.e. locally homeomorphic to separable Hilbert space. The proof was rather abstract, but it leaves open the question of whether it can just be displayed directly; it wasn't believed to be so nice back when the foundational work was being done, so I'm not sure anyone ever tried.
I guess the best way to get a direct proof is to just find the basis!
What is an explicit orthonormal basis for a neighborhood of $id \in$ Homeo$^+(\mathbb{R}^2)$?
I've been trying to do this myself using very simple functions, but haven't been able to come up with it. Here I don't want to a priori assume that the group is a Hilbert manifold, I want to prove it directly (or at least the local homeomorphism).
It seems like this is more a problem for people in functional analysis, so hopefully somebody can help!