It is sometimes useful in functional analysis to take direct limits of function spaces. For instance, the space $\mathcal{D}(\mathbf{R}^d)$ of test functions is the direct limit of the family of function spaces $C_c^\infty(K)$ in the category of locally convex topological vector spaces, i.e. such that a convex subset of $\mathcal{D}(\mathbf{R}^d)$ is open if and only if it's intersection with $C_c^\infty(K)$ is open for each compact set $K \subset \mathbf{R}^d$. We can also consider the direct limit in the family of topological vector spaces. In this sense, any (not necessarily convex) subset of $\mathcal{D}(\mathbf{R}^d)$ will be open if it's intersection with each $C_c^\infty(K)$ is open. Are these topologies different? If so, what is a set which is open in the latter topology, but not the former. If these two topologies are the same, are there other examples of direct limits where the locally convex limit differs from the general limit?
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$\begingroup$ What you say is not true: Also for the direct limit topology in the category of topological vector spaces a set with open intersections with all "steps" need not be open. The direct limit topology in the category of all topological spaces has this property. $\endgroup$– JochenFeb 18, 2022 at 9:54
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$\begingroup$ For countable systems of locally convex spaces, the direct limits in the categeory of topological vector spaces and the category of locally convex topological vector spaces coincide. $\endgroup$– JochenFeb 18, 2022 at 9:55
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$\begingroup$ Hi Jochen, thanks for your response. What would be the correct definition of a topology on the direct limit of topological vector spaces then? $\endgroup$– Jacob DensonFeb 20, 2022 at 15:10
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$\begingroup$ The finest vector space topology making all inclusions continuous. $\endgroup$– JochenFeb 21, 2022 at 11:50
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A concrete way to describe the locally convex direct limit topology: Take the collection of all the convex sets in ${\mathcal D}$ that have open intersection with all "steps" ${\mathcal D}_K$, and then close that collection up under translation, and then close that collection under arbitrary union. Compare with Definition 6.3 in Rudin's Functional Analysis, in which the direct limit is being described in the category of locally convex topological vector spaces.
I would really like to understand why, for countable systems of locally convex spaces, the direct limit in locally convex TVSs coincides with the direct limit in general TVSs. Could someone give a good reference (or a simple argument), please?
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$\begingroup$ Given a $0$-neighbourhood $U_0$ in the direct limit topology $\tau$ in the category of topological vector space, continuity of the sum yields a sequence $U_n$ of $\tau$-neighbourhoods of $0$ such that $U_n+U_n\subseteq U_{n-1}$ for all $n\in \mathbb N$. The continuity of the inclusions $X_n\to X$ and the local convexity of $X_n$ yield absolutely convex $0$-neighbourhoods $V_n$ in $X_n$ with $V_n\subseteq U_n$. Then the set $V=\sum_n V_n$ of all finite sums $v_1+\cdots+v_n$ with $v_k\in V_k$ is a $0$-neighbourhood in the locally convex direct limit topology which is contained in $U_0$ ... $\endgroup$– JochenMay 6, 2022 at 16:24
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$\begingroup$ ... since $v_1+\cdots+v_n\in v_1+\cdots v_{n-2} +U_{n-2} \subseteq v_1+\cdots+v_{n-3}+U_{n-3} \subseteq\cdots\subseteq U_1+U_1\subseteq U_0$. $\endgroup$– JochenMay 6, 2022 at 16:27