  Let $H$ be a separable real Hilbert space. Denote by ${\mathcal
G}_{\infty}(H)$ the Grassmannian consisting of closed subspaces with infinite
dimension and codimension. This Grassmannian is partially ordered by the
inclusion relation. We show that every order preserving transformation of
${\mathcal G}_{\infty}(H)$ can be extended to an automorphism of the lattice of
closed subspaces of $H$. It follows from Mackey's result \cite{Mackey} that
automorphisms of this lattice are induced by invertible bounded linear
operators.
