Proposition 9.4.2. Let $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$ be a downward-directed system of locally convex spaces over $K \in \RC$, then $E = \lim_{\longleftarrow}E_{i}$ is locally convex.
Proof. By (U) of Definition 8.9.2 and Definition 8.9.1, $E$ is equipped with the projective topology generated by the projection maps $E \to E_{i}$. By Proposition 9.4.1, $E$ is locally convex.$\square$