Characterizing weak-operator continuous linear functionals on $B(H)$ constructively

Abstract

Let $B(H)$ be the space of bounded operators on a not-necessarily-separable Hilbert space $H$. Working within Bishop-style constructive analysis, we prove that certain weak-operator continuous linear functionals on $B(H)$ are finite sums of functionals of the form $T⇝⟨Tx,y⟩$. We also prove that the identification of weak- and strong-operator continuous linear functionals on $B(H)$ cannot be established constructively.