The von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the algebraic and topological sides of operator theory.
The formal statement of the theorem is as follows. Let A be a C* algebra of bounded operators on a Hilbert space H, such that the only closed subspaces of H left invariant by every operator in A are the zero subspace and H itself. Then the closures of A in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant A'' of A. This algebra is the von Neumann algebra generated by A.
