Definition 8.10.3 (Strict). Let $(\seqi{E}, \bracsn{\iota^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of TVSs over $K \in \RC$, then the system is strict if:

1. For each $i, j \in I$ with $i \lesssim j$, $\iota^{i}_{j}: E_{i} \to E_{j}$ is injective.

2. For each $i, j \in I$ with $i \lesssim j$, the topology of $E_{i}$ is induced by $\iota^{i}_{j}$.