Geometrical characterization of semilinear isomorphisms of vector spaces and semilinear homeomorphisms of normed spaces

Abstract

Let and be vector spaces over division rings (possibly infinite-dimensional) and let and be the associated projective spaces. We say that is a PGL-mapping if for every there exists such that . We show that for every PGL-bijection, the inverse mapping is a semicollineation. Also, we obtain an analogue of this result for the projective spaces associated to normed spaces.

Keywords:

• projective space
• normed space

AMS Subject Classifications:

• 51A10
• 46B03