Constructing Extensions of Ultraweakly Continuous Linear Functionals

Let R be a linear subset of the space B(H) of bounded operators on a Hilbert space H with an orthonormal basis. It is shown constructively that if the unit ball of R is weak-operator totally bounded, then an ultraweakly continuous linear functional on R extends to one on B(H), and the extended functional has the form T↦∑∞n=1 〈Txn, yn〉, where ∑∞n=1 ‖xn‖2 and ∑∞n=1 ‖yn‖2 are convergent series in R.