The weak convergence of unit vectors to zero in the Hilbert space is the convergence of one-dimensional subspaces in the order topology

In this paper we deal with the (o)-convergence and the order topology in the hilbertian logic L ( H ) \mathcal {L}(H) of closed subspaces of a separable Hilbert space H. We compare the order topology on L ( H ) \mathcal {L}(H) with some other topologies. The main result is a theorem which asserts that the weak convergence of a sequence of unit vectors to zero in H is equivalent to the convergence of the sequence of one-dimensional subspaces generated by these vectors to the zero subspace in the order topology on L ( H ) \mathcal {L}(H) .