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.