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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 HilbertC*-modules (submitted)) we investigate analogous properties in the modulecontext. As an environment serves the class of Hilbert C*-modules. Someproperties of weak module topologies on Hilbert C*-modules are described.Natural module analogues of the classical weak Banach-Saks and the classicalSchur properties are defined and studied. A number of useful characterizationsof properties of Hilbert C*-modules is obtained. In particular, someinterrelations of these properties with the self-duality property of countablygenerated Hilbert C*-modules are established.

Module weak Banach-Saks and module Schur properties of Hilbert C*-modules