Every
infinite dimensional separable non-normable Fréchet space admits
a continuous hypercyclic operator. A large class of separable countable
inductive limits of Banach spaces with the same property is given, but
an example of a separable countable inductive limit of Banach spaces which
admits no hypercyclic operator is provided. It is also proved that
no compact operator on a locally convex space is hypercyclic.