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Let $V$ and $V'$ be vector spaces over division rings (possibleinfinite-dimensional) and let ${\mathcal P}(V)$ and ${\mathcal P}(V')$ be theassociated projective spaces. We say that $f:{\mathcal P}(V)\to {\mathcalP}(V')$ 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-bijectionthe inverse mapping is a semicollineation. Also, we obtain an analogue of thisresult for the projective spaces associated to normed spaces.

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