Chow's theorem for linear spaces

Abstract

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