Exercise 1, § 3

$a$) Show that, if a Hausdorff topological vector space E over a non-discrete valued division ring K is such that every neighbourhood of 0 contains a vector subspace that is not the single point 0, then the topology of E cannot be defined by a norm, In particular, a product of an infinite sequence $(E_n)$ of Hausdorff topological vector spaces on K, none consisting of the single point 0, has a topology that cannot be defined by a norm. b) Consider the product vector space $E = K_s^\mathbf{N}$; for all $x = (\xi_n) \in E$, put

Show that the topology of E is defined by the distance $d(x, y) = |x - y|$, that $|\lambda x| \leq |x|$ if $|\lambda| \leq 1$, $|\lambda x| \leq |\lambda|.|x|$ if $|\lambda| \geq 1$ and that, for all $x_0 \in E$, $|\lambda x_0|$ tends to 0 with $|\lambda|$.