Exercise 4, § 1

Let E be a topological vector space over a non-discrete metrizable topological field K. Show that if E is a Baire space and if in E there exists a countable base for the bornology formed by bounded subsets of E (III, p. 37, exerc. 1), then there exists a neighbourhood of 0 in E which is bounded, and consequently the Hausdorff topology associated with the topology of E is metrizable (III, p. 37, exerc. 2) (compare with exerc. 6).