Exercise 11, § 1

Let E be a topological vector space over a field K with a non discrete valuation. a) For a balanced subset A of E to absorb every bounded subset (III, p. 37, exerc. 1) of E, it is sufficient that A absorbs the set of points of every sequence $(x_n)$ tending to 0 in E. Then A is said to be bornivorous. b) Let u be a linear mapping from E into a topological vector space F over K. The image of every bounded subset of E under u is bounded in F if and only if for every sequence $(x_n)$ of points of E tending to 0, the sequence $(u(x_n))$ is bounded in F. c) Suppose E is metrizable. Show that every bornivorous subset of E is a neighbourhood of 0 in E. Deduce that if u is a linear mapping from E into a topological vector space F over K which transforms every sequence converging to 0 in E into a bounded sequence in F, then u is continuous on E.