One of the premiere mathematical structures of Quantum Mechanics is the Hilbert Space. Here, we can think of Hilbert spaces as a state space in which all vectors representing quantum states exist. The difference between these spaces, and typical vector spaces, is that Hilbert space comes equipped with an inner product, an operation that can be performed on two vectors, that returns a scalar.
[Hilbert Space] was established systematically by von Neumann, summarized in his book. The book was based on three papers, published in 1927. Of these three papers, the first introduced the notion of abstract Hilbert space, and presented the "eigenvalue problem", of self-adjoint operators having a continuous part in their spectrum, in a mathematically rigorous form, without making use of Dirac's delta function. The complete, analysis of the spectral theorem was worked out in a following paper in 1930.
The surface of this sphere is a valid Hilbert Space, along with the inner product between qubit states. A final note on Hilbert spaces is that they relate to Unitary matrices, which preserve inner product, meaning you can transform a vector under a series of unitary matrices without breaking the normalization condition that the inner product of two vectors must equate to one: