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von Neumann bicommutant theorem

In mathematics, specifically functional analysis, 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 : Von Neumann bicommutant theorem. Let $M$ be an algebra consisting of bounded operators on a Hilbert space H, containing the identity operator, and closed under taking adjoints. Then the closures of $M$ in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant $M''$ of $M$ .
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem)

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