Hilbert -modules are generalizations of Hilbert spaces, but the theory of Hilbert -modules is different from the theory of Hilbert spaces; for example, not all closed submodules of a given Hilbert -modules are orthogonally complemented. It is well-known that the closed unit ball of a Hilbert space is weakly sequentially compact. This result is not true for Hilbert -modules.
A sequence in a Hilbert -module over a -algebra is weakly convergent to an element if the sequence converges to with respect to the -norm on , for each . The authors prove that the closed unit ball of a full Hilbert -module over a -algebra is weakly sequentially compact if and only if the -algebra is finite-dimensional.