Exercise 1, § 1

Let $E_0 = \mathbf{Q}_p^\mathbf{N}$ be the vector space over the $p$-adic field $\mathbf{Q}_p$ (GT, III, § 6, exerc. 23) which is the product of an enumerable infinity of factor each identical with $\mathbf{Q}_p$. Let $P \subset E_0$ be the set $\mathbf{Z}_p^\mathbf{N}$, and let $E$ be the vector subspace of $E_0$ generated by $P$. On the additive group $P$ we consider the product compact topology of the topologies of the factors $\mathbf{Z}_p$, and we denote by $\mathfrak{B}$ the filter of neighbourhoods of 0 in $P$ for this topology. Show that $\mathfrak{B}$ is a fundamental system of neighbourhoods of 0 in $E$ for a topology $\mathscr{T}$ compatible with the additive group structure of $E$, that satisfies (EVT'_I) and (EVT'_III) but not (EVT'_II) (prove that the homothety $x \mapsto x/p$ is not continuous in $E$).