Lemma

— Let $M$ and $N$ be two vector spaces, and $g$ a linear mapping of $M$ in $N$. If $\mathscr{T}_0$ is a topology compatible with the vector space structure of $N$, then the inverse image of $\mathscr{T}_0$ by $g$ is compatible with the vector space structure of $M$.

We show, for example, that $(\lambda, x) \mapsto \lambda x$ is continuous at each point $(\lambda_0, x_0)$ of $K \times M$. Put $y_0 = g(x_0)$. Every neighbourhood of 0 in $M$ contains a neighbourhood of the form $g(U)$ where $U$ is a neighbourhood of 0 in $N$; by hypothesis there exists a neighbourhood $V$ of 0 in $K$ and a neighbourhood $W$ of 0 in $N$ such that the relations $\lambda - \lambda_0 \in V$, and $y - y_0 \in W$ imply $\lambda y - \lambda_0 y_0 \in U$. Thus the relations $\lambda - \lambda_0 \in V$, $x - x_0 \in g(W)$ imply $\lambda x - \lambda_0 x_0 \in g(U)$. We can show similarly that $(x, y) \mapsto x - y$ is continuous in $M \times M$.

For each index $i \in I$, let $\mathscr{B}_i$ be a fundamental system of neighbourhoods of 0 in $E_i$. From the definition of the topology $\mathscr{T}$, the filter of neighbourhoods of 0 for this topology is generated by unions of sets of the families $f_i^{-1}(\mathscr{B}_i)$; in other words, the sets of the form $\bigcap_k f_{i_k}^{-1}(V_{i_k})$ form a fundamental system of neighbourhoods of 0 for $\mathscr{T}$, the $(i_k)_{1 \leq k \leq n}$ being any finite sequence of indices of I, and, for each index $k$, $V_{i_k}$ any set of $\mathscr{B}_{i_k}$.