In projective geometry, a correlation is a transformation of a d-dimensional projective space that maps subspaces of dimension k to subspaces of dimension d − k − 1, reversing inclusion and preserving incidence.
In general n-dimensional projective space, a correlation takes a point to a hyperplane.
This context was described by Paul Yale: He proves a theorem stating that a correlation φ interchanges joins and intersections, and for any projective subspace W of P(V), the dimension of the image of W under φ is (n − 1) − dim W, where n is the dimension of the vector space V used to produce the projective space P(V).
There is a natural correlation induced between a projective space P(V) and its dual P(V∗) by the natural pairing ⟨⋅,⋅⟩ between the underlying vector spaces V and its dual V∗, where every subspace W of V∗ is mapped to its orthogonal complement W⊥ in V, defined as W⊥ = {v ∈ V | ⟨w, v⟩ = 0, ∀w ∈ W}.
In this way, every nondegenerate semilinear map V → V∗ induces a correlation of a projective space to itself.