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

Abstract

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