Exercise 1, § 1

Let E be a left topological vector space over a non-discrete topological field K. A subset B of E is said to be bounded if for every neighbourhood V of 0 in E there exists $\lambda \neq 0$ in K such that $\lambda B \subset V$. a) Show that if B is bounded in E, then for every neighbourhood V of 0 in E, there exists a neighbourhood U of 0 in K such that $U.B \subset V$. b) Show that the closure of a bounded set in E is bounded. The union of two bounded sets is bounded. Every precompact set in E is bounded. Extend the corollaries of III, p. 4, prop. 4 to topological vector spaces over K. c) Prove that if A is a bounded set in K (considered as a vector space on the left over itself) and B is a bounded set in E, then A.B is bounded in E. d) Extend prop. 3 of III, p. 4 to the case where K is a metrizable topological division ring. e) Extend the notion of a quasi-complete space and its properties to topological vector spaces.