Just a quick question about the geometry of Hilbert spaces from an intuitive standpoint. Maybe just assuming we're working with $L^2$ would simplify the situation. Basically, in something like $\mathbb{R}^2$ we have the situation that $\cos(\theta)=\frac{\langle a,b\rangle}{\vert a\vert\cdot\vert b\vert}$, and the idea of an angle between vectors is very meaningful, geometrically. We can easily extend this idea to $\mathbb{R}^n$ because when we talk about the angle between two vectors, we mean we are choosing the plane that both of them lie in, and picking the vector in there. But what does this really mean in a Hilbert space like $L^2$? I have a good intuition about functions, and about geometry (topology) separately, but not really the "geometry of functions".
Now, there may be no visualization of this in $L^2$, and I'm not asking for one, but is there any sense to doing geometry (i.e. actual polygons, things like that) in a space like $L^2$? Also, what sort of applications do ideas like this have in functional analysis? Are we ever interested in ideas like "planes" of functions, polygons, surfaces, solids, etc.? What do we really mean by angles, projections, normal vectors? And do these sorts of things ever have any sort of interesting relationships?
I'm primarily asking for an intuitive idea here. It's easy to just do the math, prove theorems about inner products, norms, etc. Maybe geometry gives some clues or intuitive ideas when doing functional analysis?