What is Banach space projective geometry?

[MichaelMohan]

What are the foundational principles and mathematical frameworks underlying the theory of projective geometry when extended to Banach spaces? How does this extension affect classical concepts such as lines, planes, and transformations? What are the key differences and similarities between projective geometry over Euclidean spaces and projective geometry over Banach spaces, and how do these differences influence the study and application of geometric structures in various mathematical contexts?

[janusz55]

In classical projective geometry , neither the distances nor the parallelism of the lines are preserved.
As you can guess, in classical projective geometry there is no place for Tales' theorem or Pythagoras' theorem. On the other hand, we see that lines remain lines and points remain points. A projective plane, on the other hand, can be thought of as an ordinary plane to which points are attached at infinity.
Somewhat more formally: consider the plane z = 0 (that is, all points of the form (x, y, 0)) and denote it by π. The point S = (0, 0, 1) lies above this plane.

In Banach-Hilbert projective geometry, the distance of the points, the parallelism of the lines, is preserved.
The projective plane can be thought of as the set of all straight lines passing through S.
We describe the projective plane as the transformation φ(x, y, z) = (xy, yz, zx, x^2 - y^2)(from the space R^3 in R ^4), which we restrict to only the points of the sphere of radius 1 and centre (0, 0, 0) - so the points for which x^2 + y^2 + z^2 = 1.
Any straight line passing through the point (0, 0, 0) pierces the described sphere at two points. If one of these is (a, b, c), then the other is (-a, -b, -c).
Thus, one can equate a line passing through the points(0, 0, 0) and (x, y, z) with the point φ(x, y, z) (lying in a four-dimensional space).

[MichaelMohan]

It seems like you are asking for uses of projective spaces which are not just to illustrate examples. There are plenty: infinite projective space represents cohomology and so the topology of projective space informs are understanding of cohomology.

[janusz55]

Projective space - a modification of geometric space by attaching all the directions of that space to the set of points of the space[. In such an augmented space, any two different projective lines lying on the same projective plane have a common proper or improper point called the point at infinity.