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

TL;DR

This paper characterizes semilinear isomorphisms of vector spaces and semilinear homeomorphisms of normed spaces through geometrical properties of their associated projective spaces, extending classical results.

Contribution

It provides a new characterization of semilinear isomorphisms and homeomorphisms via projective space mappings, generalizing known geometric theorems.

Findings

PGL-bijections have inverse mappings that are semicollineations

Results extend to projective spaces of normed spaces

Characterization of semilinear isomorphisms through geometric properties

Abstract

Let $V$ and $V'$ be vector spaces over division rings (possible infinite-dimensional) and let ${\mathcal P}(V)$ and ${\mathcal P}(V')$ be the associated projective spaces. We say that $f:{\mathcal P}(V)\to {\mathcal P}(V')$ is a PGL-{\it mapping} if for every $h\in {\rm PGL}(V)$ there exists $h'\in {\rm PGL}(V')$ such that $fh=h'f$. 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.