Proposition 4

*In a topological vector space E on a non-discrete valued division ring K there exists a fundamental system $\mathfrak{B}$ of closed neighbourhoods of 0 such that : (EV₁) Every set V ∈ $\mathfrak{B}$ is balanced and absorbent. (EV₂) For every V ∈ $\mathfrak{B}$ and λ ≠ 0 in K, we have λV ∈ $\mathfrak{B}$ (invariance of $\mathfrak{B}$ under homotheties of non zero ratio). (EV₃) For every V ∈ $\mathfrak{B}$, there exists W ∈ $\mathfrak{B}$ such that W + W ⊂ V.

Conversely, let E be a vector space on K, and let $\mathfrak{B}$ be a filter base on E satisfying the conditions (EV₁), (EV₂) and (EV₃). Then there exists a topology (and it is unique) on E, compatible with the vector space structure of E, and for which $\mathfrak{B}$ is a fundamental system of neighbourhoods of 0.

By axiom (EVT'₃) we show firstly that the balanced core, V₁, of V, a neighbourhood of 0, is itself a neighbourhood of 0. For there exist α > 0 and a neighbourhood W of 0 such that, if |λ| ≤ α and x ∈ W, then λx ∈ V. Since K is non-discrete, there exists μ ≠ 0 in K with |μ| ≤ α and μW is a neighbourhood of 0 for which μW ⊂ V. Also if v ∈ K and |v| ≤ 1 then |vμ| ≤ α and thus vμW ⊃ V. Hence μW ⊃ V₁ and V₁ is a neighbourhood of 0. Also as V is closed so also V₁ is closed. Thus the set $\mathfrak{B}$ of closed balanced neighbourhoods of 0 form a fundamental system of neighbourhoods of 0 in E. By (EVT₁') every neighbourhood of 0 is absorbent; furthermore $\mathfrak{B}$ satisfies (EV₂) (cf. I, p. 3, cor.); finally, because of the continuity of (x, y) ↦ x + y at the point (0, 0), every fundamental system of neighbourhoods of 0 in E satisfies (EV₃). The set $\mathfrak{B}$ satisfies the conditions of the proposition.

Now let E be a vector space over K, and $\mathfrak{B}$ be a filter base on E satisfying (EV₁), (EV₂) and (EV₃). The axiom (EV₁) shows firstly that for all V ∈ $\mathfrak{B}$, we have −V = V and 0 ∈ V; these relations and the axiom (EV₃) show that $\mathfrak{B}$ is a fundamental system of neighbourhoods of 0, for a topology on E compatible with the additive group structure of E (GT, III, § 1.2). On the other hand the axioms (EVT'_I), (EVT'_II) and (EVT'_III) are immediate consequences of (EV_I) and (EV_II), thus the topology defined above satisfies the axiom (EVT), and the proposition is proved.