Definition 8.9.2 (Projective Limit of Topological Vector Spaces). Let $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$ be a downward-directed system of topological vector spaces over $K \in \RC$, then there exists $(E, \bracsn{T^E_i}_{i \in I})$ such that:
$E$ is a TVS over $K$.
For each $i \in I$, $T^{E}_{i} \in L(E; E_{i})$.
For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
\[\xymatrix{ E_i \ar@{->}[r]^{T^i_j} & E_j \\ E \ar@{->}[u]^{T^E_i} \ar@{->}[ru]_{T^E_j} & }\]For any pair $(F, \bracsn{S^F_i}_{i \in I})$ satisfying (1), (2), and (3), there exists a unique $S \in L(F; E)$ such that the following diagram commutes
\[\xymatrix{ & E_i \\ F \ar@{->}[r]_{S} \ar@{->}[ru]^{S^F_i} & A \ar@{->}[u]_{T^E_i} }\]for all $i \in I$.
For any TVS $F$ over $K$ and $S \in \hom(F; E)$, $S \in L(F; E)$ if and only if $T^{E}_{i} \circ S \in L(F; E_{i})$ for all $i \in I$.
The pair $(E, \bracsn{T^E_i}_{i \in I})$ is the projective limit of $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$.
Proof. Let $(E, \bracsn{T^E_i}_{i \in I})$ be the inverse limit of $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$ as $K$-vector spaces (Proposition 1.2.10).
Equip $E$ with the projective topology generated by $\bracsn{T^E_i}_{i \in I}$, then $(E, \bracsn{T^E_i}_{i \in I})$ satisfies (1), (2), and (3).
(5): By (5) of Definition 8.9.1.
(U): By (U) of Proposition 1.2.10, there exists a unique $S \in \hom(F; E)$ such that the given diagram commutes. By (4), $S \in L(F; E)$.$\square$