Let H be an infinite-dimensional Hilbert space of density character m. By representing H as a module over an appropriate Clifford algebra, it is proved that H possesses a family {Aα}α∈m of proper closed nonzero subspaces such that Analogous results are proved for Lp spaces and for c 0 (X) and p(X) (1 ≤ p ≤ ∞) when X is an arbitrary nonzero Banach space.