Remark

— In the last example, the topology induced by $\mathscr{T}$ on $E_K$ is identical with $\mathscr{T}_K$, since by definition it is coarser than $\mathscr{T}_K$ and, since $\mathscr{T}$ is finer than $\mathscr{T}_u$, the topology induced by $\mathscr{T}$ on $E_K$ is finer than that induced by $\mathscr{T}_u$, that is to say $\mathscr{T}_K$.

This reasoning generalises immediately to an inductive limit of locally convex topologies $(\mathscr{T}_\alpha)$ when there is a locally convex topology $\mathscr{T}'$ on E such that $\mathscr{T}_\alpha$ is the topology induced on $E_\alpha$ by $\mathscr{T}'$.

More generally one can ask, when we suppose that $E_\beta \subset E_\alpha$ and $\mathscr{T}_\beta$ is the topology induced by $\mathscr{T}_\alpha$, under what circumstances $\mathscr{T}$ induces $\mathscr{T}_\alpha$ on each of the $E_\alpha$. In general this is not so (II, p. 80, exerc. 26); but we shall see in the Nos following two important situations where this does occur.