Past exam of the analysis course of the University of Oxford / 2010 / B / 5 / vi / Solution

By the Cauchy-Schwarz inequality for a positive operator, for every $y$,Taking $y=Tx$ gives $Tx=0$, and $(S+T)x=0$ then gives $Sx=0$. ThusThe reverse inclusion is immediate. This is the kernel of a sum of positive operators identity.