Exercise 14, § 1

Show that if the topology of a metrizable locally convex space E cannot be defined by a single norm, then there does not exist a countable base for the canonical bornology of E (using III, p. 38, exerc. 5, show that otherwise there will exist a bounded bornivorous set (III, p. 39, exerc. 11) in E, and complete the argument using III, p. 39, exerc. 11, c)).