Chow's Theorem for Linear Spaces

Abstract

If $\varphi :L\to L^{\prime }$ is a bijection from the set of lines of a linear space $(P,L)$ onto the set of lines of a linear space $(P^{\prime },L^{\prime })$ ($\dim P,\dim P^{\prime }\ge 3$), such that intersecting lines go over to intersecting lines in both directions, then $\varphi$ is arising from a collineation of $(P,L)$ onto $(P^{\prime },L^{\prime })$ or a collineation of $(P,L)$ onto the dual linear space of $(P^{\prime },L^{\prime })$. However, the second possibility can only occur when $(P,L)$ and $(P^{\prime },L^{\prime })$ are 3-dimensional generalized projective spaces.