Remark 1

If B is a precompact set of K, then for every neighbourhood V of 0 in E, there is a neighbourhood U of 0 in E such that BU $\subset$ V. For, if W is a neighbourhood of 0 in E such that W + W $\subset$ V; then from (EVT$_{\text{III}}$) there is a neighbourhood T$_0$ of 0 in K and a neighbourhood U$_0$ of 0 in E such that T$_0$U$_0$ $\subset$ W. As B is precompact, there are finitely many points $\lambda_i \in B$ ($1 \leq i \leq n$) such that the $\lambda_i + T_0$ cover B; from (EVT$_{\text{II}}$) it follows that there is a neighbourhood U $\subset$ U$_0$ of 0 in E, such that $\lambda_i U \subset W$ for all $i$; clearly U has the required properties. In a similar manner (using (EVT$_{\text{I}}$) instead of (EVT$_{\text{II}}$)) it can be shown that if H is a precompact set of E, then for every neighbourhood V of 0 in E, there exists a neighbourhood T of 0 in K such that TH $\subset$ V.