In mathematics and the foundations of quantum mechanics, the projective Hilbert space P {\displaystyle \mathbf {P} } of a complex Hilbert space H {\displaystyle H} is the set of equivalence classes of non-zero vectors v {\displaystyle v} in H {\displaystyle H}, for the relation ∼ {\displaystyle \sim } on H {\displaystyle H} given by w ∼ v {\displaystyle w\sim v} if and only if v = λ w {\displaystyle v=\lambda w} for some non-zero complex number λ {\displaystyle \lambda }. Wikipedia