 Von Neumann double commutant theorem

For a unital star-subalgebra $A$ of the bounded operators on a complex Hilbert space, its strong operator topology and weak operator topology closures equal its double commutant of an operator algebra. To approximate $T\in A^{\prime \prime }$ on finitely many vectors, take their tuple in $H^m$. The projection onto its diagonal-$A$ cyclic subspace has matrix entries in $A^{\prime }$, so it commutes with the diagonal action of $T$. Therefore the image tuple under $T$ lies in that cyclic subspace and can be approximated by a single $a\in A$.