Direct limits of topological vector spaces

Question

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?

Answer 1

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?