isometries from H to G and from G to H. Fix two such partial isometries for the argument. One has In turn, By induction, and Now each additional summand in the direct sum expression is obtained using one of the two fixed partial isometries, so This proves the theorem. See also Schröder–Bernstein theorem for measurable spaces Schröder–Bernstein property References B. Blackadar, Operator Algebras, Springer, 2006. C*-algebras Functional analysis Operator theory Von Neumann algebras