Final topology coinduced by topological vector spaces

Question

Let $X$ be a vector space, $(X_i)$ a collection of (not necessarily locally convex) topological vector spaces, and $T_i \colon X_i \to X$ linear maps. Then the $T_i$ coinduce a final topology $\mathcal{T}_1$ on $X$, i.e., the finest topology on $X$ with respect to which the $T_i$ are continuous.

On the other hand, the lattice of vector topologies on $X$ is join-complete (since the initial topology induced by topological vector spaces is a vector topology), hence complete, so there is a finest vector topology $\mathcal{T}_2$ such that the $T_i$ are continuous.

Of course $\mathcal{T}_2 \subseteq \mathcal{T}_1$, and we do not in general have equality. (For instance, if there is only one $X_i$ and $T_i$ is not surjective, then $T_i[X_i]$ is clopen in $\mathcal{T}_1$.) But what assumptions are sufficient to ensure equality?

If the $X_i$ are locally convex then we can do something similar (since the lattice of locally convex topologies on $X$ is also join-complete) and obtain a finest locally convex topology $\mathcal{T}_3 \subseteq \mathcal{T}_2$. When do we have equality between the $\mathcal{T}_j$ in this case?

Answer 1

As an example of sufficient conditions, check Bourbaki, "Topological vector spaces", ch 3, sec 1.4. There is a proposition:

Let $E_n$ be a sequence of Hausdorff locally convex spaces, and for every $n,$ let $u_n: E_n \to E_{n + 1}$ be an injective linear mapping which is compact. Let $E = \varinjlim_{\mathrm{lc}} E_n$ be the inductive limit of the system $(E_n, u_n),$ and let $v_n$ be the canonical mapping from $E_n$ into E. Then the locally convex space $E$ is Hausdorff. Moreover, for every subset $A$ of $E,$ the following conditions are equivalent:

1. $A$ is bounded;
2. there exists an integer $n$ such that $A$ is the image under $v_n$ of a bounded subset of $E_n;$
3. $A$ is relatively compact.

and lemma

Under the hypothesis of previous proposition, the topology of $E$ is the finest topology for which all the mappings $v_n: E_n \to E$ are continuous.

That is inductive limit in category of locally convex spaces and in category of topological spaces are the same.