Let be a barrelled TVS and be a locally convex TVS.
Let be a subset of the space of continuous linear maps from into .
The following are equivalent:
is bounded for the topology of pointwise convergence;
is bounded for the topology of bounded convergence;
is equicontinuous. (en)
If is a barrelled TVS over the complex numbers and is a subset of the continuous dual space of , then the following are equivalent:
is weakly bounded;
is strongly bounded;
is equicontinuous;
is relatively compact in the weak dual topology. (en)
Every closed linear operator from a Hausdorff barrelled TVS into a complete metrizable TVS is continuous. (en)
