In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector formula_1 belonging to a (separable) complex Hilbert space formula_2. This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys formula_3, and it is well-defined up to a complex number of modulus 1 (the global phase), that is, formula_1 and formula_5 represent the same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions formula_6, while the Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors formula_7 with the usual inner product.