Apartments preserving transformations of Grassmannians of infinite-dimensional vector spaces

We define the Grassmannians of an infinite-dimensional vector space V as the orbits of the action of the general linear group GL(V) on the set of all subspaces. Let G be one of these Grassmannians. An apartment in G is the set of all elements of G spanned by subsets of a certain basis of V. We show that every bijective transformation f of G such that f and f−1 send apartments to apartments is induced by a semilinear automorphism of V. In the case when G consists of subspaces whose dimension and codimension both are infinite, a result of such kind will be proved also for the connected components of the associated Grassmann graph.