  Given a complex structure $J$ on a real (finite or infinite dimensional)
Hilbert space $H$, we study the geometry of the Lagrangian Grassmannian
$\Lambda(H)$ of $H$, i.e. the set of closed linear subspaces $L\subset H$ such
that $$J(L)=L^\perp.$$ The complex unitary group $U(H_J)$, consisting of the
elements of the orthogonal group of $H$ which are complex linear for the given
complex structure, acts transitively on $\Lambda(H)$ and induces a natural
linear connection in $\Lambda(H)$. It is shown that any pair of Lagrangian
subspaces can be joined by a geodesic of this connection. A Finsler metric can
also be introduced, if one regards subspaces $L$ as projections $p_L$ (=the
orthogonal projection onto $L$) or symmetries $\e_L=2p_L-I$, namely measuring
tangent vectors with the operator norm. We show that for this metric the
Hopf-Rinow theorem is valid in $\Lambda(H)$: a geodesic joining a pair of
Lagrangian subspaces can be chosen to be of minimal length. We extend these
results to the classical Banach-Lie groups of Schatten.
