Remark 3

If K is a discrete division ring, conditions (EVT'_I) and (EVT'_III) are true for any topology on E. Arguing as in prop. 4, one easily sees that if E is a topological vector space on K, then there exists $\mathfrak{B}$, a fundamental system of closed neighbourhoods of 0 in E satisfying conditions (EV_II) and (EV_III). Conversely, if a filter base $\mathfrak{B}$ on a vector space E over K is such that 0 belongs to all the sets of $\mathfrak{B}$ and (EV_II), (EV_III) are true, then $\mathfrak{B}$ is a fundamental system of neighbourhoods of 0 in a topology compatible with the vector space structure of E.