We obtain a counterexample about the hypothesis of a linear operator 
on a locally convex space to a Hilbert space mapping bounded convergent 
nets to convergent nets, implying that the operator is continuous. 
Accordingly, a linear operator defined on a locally convex space 
and, taking values in a Hilbert space, might map summable families to 
summable families without being continuous. 
We may take the weak dual of a Hilbert space for the domain space.