Dieudonné's theorem

Dieudonné's theorem states that every collineation between projective spaces of dimension at least two is induced by a semilinear transformation between their underlying vector spaces.

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Projective space: A geometric space whose points are one-dimensional subspaces of a vector space. The theorem describes which maps between these spaces count as collineations.
Vector space: A set equipped with addition and scalar multiplication satisfying the vector space axioms. The theorem lifts projective geometry back to transformations of these underlying spaces.
Jean Dieudonné: A French mathematician known for work in algebra, functional analysis, and the history of mathematics. The theorem bears his name and appears in his treatment of projective geometry.
Linear transformation: A map between vector spaces that preserves vector addition and scalar multiplication. It induces a collineation only when no nontrivial field automorphism is needed.
Fundamental theorem of affine geometry: A theorem stating that suitable bijections preserving affine lines arise from semiaffine transformations. Its affine analogue uses a similar semilinear conclusion for line-preserving maps.
Collineation: A bijection between projective spaces that maps lines to lines. This is the geometric transformation the theorem characterizes.
Projective line: The projective space formed by the one-dimensional subspaces of a two-dimensional vector space. The dimension restriction excludes this case, where line-preserving bijections need not be semilinear.
Projective geometry: The geometry of points and lines defined by vector spaces, where perspective projection preserves incidence. The theorem arose from classifying transformations intrinsic to this geometry.
Projective duality: A correspondence interchanging points and hyperplanes while reversing incidence in projective geometry. Unlike a collineation, a duality need not map points to points.
Automorphism group: The group of structure-preserving bijections from a mathematical object to itself. Dieudonné's theorem identifies the algebraic form of projective-space automorphisms.