Exercise 2, § 1

a) Let E be a left topological vector space over a non-discrete topological field K. Prove that if there exists a neighbourhood V of 0 in E which is bounded (exerc. 1), than the sets $\lambda V$, for $\lambda \in K$ and $\lambda \neq 0$, form a fundamental system of neighbourhoods of 0 in E. If K is metrizable, the Hausdorff topology associated with the topology of E (GT, III, § 2, No. 6) is metrizable. If $K = \mathbf{R}$ or $K = \mathbf{C}$, the locally convex topology on E which is the finest of the topologies coarser than the given topology on E (II, p. 80, exerc. 23) can be defined by a single semi-norm. b) Prove that the topology of an infinite product of locally convex Hausdorff spaces (of which none is just 0) cannot be defined by a single semi-norm. c) Let E be a locally convex space whose topology is defined by an increasing sequence $(p_n)$ of semi-norms. In order that the topology of E be defined by a single semi-norm, it is necessary and sufficient that there exists an integer $n_0$ such that for every $n \geq n_0$ there exists a number $k_n \geq 0$ such that $p_n(x) \leq k_n p_{n_0}(x)$ for all $x \in E$. d) Let E be the vector space over $\mathbf{R}$ consisting of infinitely differentiable numerical functions on the internal $I = (0, 1)$. For every integer $n \geq 0$, let $p_n(f) = \sup_{0 \leq k \leq n} (\sup_{x \in I} |f^{(k)}(x)|)$ (with f^{(0)} = f); show that the $p_n$ are norms on E and that the topology defined by the sequence of norms $p_n$ cannot be defined by a single norm.