= Von Neumann double commutant theorem
{c}
{title2=$\overline A^{\mathrm{SOT}}=\overline A^{\mathrm{WOT}}=A''$}
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 $T\in A''$ on finitely many vectors, take their tuple in $H^m$. The projection onto its diagonal-$A$ cyclic subspace has matrix entries in $A'$, 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 $a\in A$.
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