Exercise 6, § 4

A locally convex space E is said to be relatively bounded if there exists a bounded barrel in E. a) In order that E be relatively bounded, it is necessary and sufficient that the topology of E is coarser than a topology defined by a semi-norm. Then there exists a base for the canonical bornology of E consisting of barrels. b) For E to be bornological and relatively bounded, it is necessary and sufficient that the topology of E is the lower bound of a family of normed space topologies on E (cf. III, p. 40, exerc. 1). Further, in order that there exist also a countable base for the canonical bornology of E, it is necessary and sufficient that the topology of E is the lower bound of a sequence of normed space topologies.