I am an amateur mathematician learning new things.
Let A and B be vectors in a Hilbert space. The three vectors A, B and A-B form a triangle. The idea of the angle between two vectors can be captured using the inner product: the arc cosine of the inner product of A and B divided by the product of the norms of A and B is the angle between A and B. Likewise, the angles between A and B-A and B and B-A can be so defined. If the Hilbert space is an n dimensional Euclidean space and the inner product is the dot product, then the sum of the three angles of the triangle will be pi. Is it possible to define an inner product so that the sum of the three angles will not be pi for an n dimensional Euclidean space or for an infinite dimensional function space? Can Hilbert spaces generalize non-Euclidean geometries or just Euclidean geometry?