Exercise 12, § 1

Let $E$ be a locally convex Hausdorff space, $E'$ its dual. a) In order that there exists a weakly bounded and absorbent set in $E'$, it is necessary and sufficient that the topology of $E$ is coarser than a normed space topology (\emph{cf.} IV, p. 47, exerc. 2). b) In order that there exists an equicontinuous and weakly total set in $E'$, it is necessary and sufficient that the topology of $E$ is finer than a normed space topology. c) In order that there exists an equicontinuous absorbent set in $E'$, it is necessary and sufficient that the topology of $E$ can be defined by a single norm.