  Continuing the research on the Banach-Saks and Schur properties started in
(cf. M. Frank, A. A. Pavlov, Banach-Saks properties of C*-algebras and Hilbert
C*-modules (submitted)) we investigate analogous properties in the module
context. As an environment serves the class of Hilbert C*-modules. Some
properties of weak module topologies on Hilbert C*-modules are described.
Natural module analogues of the classical weak Banach-Saks and the classical
Schur properties are defined and studied. A number of useful characterizations
of properties of Hilbert C*-modules is obtained. In particular, some
interrelations of these properties with the self-duality property of countably
generated Hilbert C*-modules are established.
