Proposition 7

Let $(E_v)_{v \in I}$ be a family of topological vector spaces on a topological division ring $K$. Let $E$ be a vector space on $K$ and for each $v \in I$, let $f_v$ be a linear mapping of $E$ in $E_v$. Then the coarsest topology on $E$ which makes each function $f_v$ continuous, is a topology $\mathscr{T}$ compatible with the vector space structure of $E$. Further, if for every $x \in E$, $\phi(x)$ denotes the point $(f_v(x))$ of the product space $F = \prod_{v \in I} E_v$, then the topology $\mathscr{T}$ is the inverse image of the topology of the subspace $\phi(E)$ of $F$ under the linear mapping $\phi$.

The last part of the proposition is a particular case of GT, I, § 4.1, prop. 3. The proposition then follows from the next lemma.