I know homeomorphisms are something like transformations of a space. But I am struggling to make some "system" of all homeomorphisms of given space (say $l^2$) and whether we can describe them all somehow.
How do the homeomorphisms on $l^2$ look like? Can they be classified or categorized somehow? Can we define a category of homeomorphisms on $l^2$? (By $l^2$, I mean the Hilbert space of all square summable sequences.)
Since $l^2$ is of similar nature to $\mathbb{R}^2$, do also the homeomorphisms on $\mathbb{R}^2$ and $l^2$ have something in common? (E.g. can we describe homeomorphisms on $l^2$ using homeomorphisms on $\mathbb{R}^2$?