I'm interested in learning how the geometry of the euclidean plane, with its problems concerning points, lines, circles, conics or higher curves, can or cannot be extended to infinite dimensions.
I understand that there are infinite-dimensional sequence-spaces and function-spaces, but the books of introductory functional analysis i know of seem to deal with different kinds of problems.
To be more specific, here are some examples concerning what i'm interested in:
In 1,2,3,4,... and all finite dimensions, the point (1,1,...,1) is well-defined. In infinite dimensions, its length is infinite, so does the point exist at all?
Also, finite dimensions also mean discrete dimensions, so we can visualize coordinate axes, up to 3, and even imagine that axes do exist for higher countable dimensions. But in spaces of real functions where dimensions are uncountable, does the concept of axis and direction still apply?
Hoping to make clearer what i'm asking for:
Triangles, circles, conics, cubes, spheres, are they useful concepts in a function space?
Also, constructions with ruler and compass?
Maybe of "platonic solids" in infinite dimensions?
Are there books that present a coherent approach along these lines?