  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.
