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

Set $B(x,y)=⟨Tx,y⟩$. It is a positive semidefinite Hermitian form. Write $A=B(x,x)$, $C=B(y,y)$ and $c=B(x,y)$. For every complex $t$,If $C>0$, choosing $t=-c/C$ gives $A-|c{|}^2/C\ge 0$. If $C=0$ and $c\ne 0$, choosing $t=-sc$ for large positive $s$ makes the expression negative, a contradiction. Thus $c=0$ in that case. In all cases,This is the Cauchy-Schwarz inequality for a positive operator.