Definition 8.10.2 (Inductive Limit). Let $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of TVSs over $K \in \RC$, then there exists $(E, \bracsn{T^i_E}_{i \in I})$ such that:
$E$ is a TVS over $K$.
For each $i \in I$, $T^{i}_{E} \in L({E_i, E})$.
For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
\[\xymatrix{ E_i \ar@{->}[rd]_{T^i_E} \ar@{->}[r]^{T^i_j} & E_j \ar@{->}[d]^{T^j_E} \\ & E }\]For any pair $(F, \bracsn{S^i_F}_{i \in I})$ satisfying (1), (2), and (3), there exists a unique $S \in L({E, F})$ such that the following diagram commutes
\[\xymatrix{ E_i \ar@{->}[d]_{T^i_E} \ar@{->}[rd]^{S^i_F} & \\ E \ar@{->}[r]_{S} & F }\]for all $i \in I$.
For any TVS $F$ and $T \in \hom(E; F)$, $T \in L(E; F)$ if and only if $T \circ T^{i}_{E} \in L(E_{i}; F)$ for all $i \in I$.
The pair $(E, \bracsn{T^i_E}_{i \in I})$ is the inductive limit of $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$.
Proof. Let $(E, \bracsn{T^i_E}_{i \in I})$ be the direct limit of $(\seqi{E}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ as vector spaces over $K$ (Proposition 1.2.9). Equip $E$ with the inductive topology induced by $\bracsn{T^i_E}_{i \in I}$, then $(E, \bracsn{T^i_E}_{i \in I})$ satisfies (1), (2), and (3).
(U): By (U) of Proposition 1.2.9, there exists a unique $S \in \hom(E; F)$ such that the given diagram commutes. By (4) of Definition 8.10.1, $S \in L(E; F)$.
(5): By (5) of Definition 8.10.1.$\square$