Proposition 9.4.1. Let $E$ be a vector space over $K \in \RC$, $\seqi{F}$ be locally convex spaces over $K$, and $\seqi{T}$ where $T_{i} \in \hom(E; F_{i})$ for all $i \in I$, then the projective topology on $E$ is locally convex.
Proof. By Definition 8.9.1,
\[\mathcal{B}= \bracs{\bigcap_{j \in J}T_j^{-1}(U_j) \bigg | J \subset I \text{ finite}, U_j \in \cn_{F_j}(0)}\]
is a fundamental system of neighbourhoods at $0$. For each $i \in I$, $U_{i} \in \cn_{F_i}(0)$ convex, $T^{-1}(U_{i})$ is also convex. Since each $F_{i}$ is locally convex, $\mathcal{B}$ contains a fundamental system of neighbourhoods at $0$ consisting of only convex sets.$\square$