Do colimits in the category of (not necessarily locally convex) topological vector spaces (over R, C, respectively) exist in general?


If no, is there a well-known condition of when they exist?


If yes, how can I describe the topology of the colimit?


(The example in my mind has following properties. First, it is a filtered colimit; it is a chain of inclusions, not necessarily linearly ordered, and not necessarily countable. Also, each inclusion is in general not a homeomorphism onto the image. Topology gets coarser and coarser as you embed into a larger space.)


(I'm also interested on a condition when a subset of the colimit is contained in one of the component spaces. For example, in the test function space $\mathcal{C}\_c^\infty(\mathbb{R}^n)$ (which is of course NOT a colimit in the category of topological vector spaces), any bounded set is contained in one of $\mathcal{C}^{\infty}(K)$, $K$ a compact set. When can I say something similar?)



Colimits in the category of (not necessarily locally convex) topological vector spaces
The Springer Lecture Notes 639 *Topological Vector Spaces* of Adasch, Ernst, and Keim contain in § 4 a more or less explicit construction of inductive (=co-) limits in the category of topological vector spaces based on the notion of a *string*: A sequence $(U\_n)\_{n\in\mathbb N}$ of balanced and absorbing sets such that $U\_{n+1}+U\_{n+1}\subseteq U\_n$. Similarly to the locally convex theory (if $U$ is absolutely convex the sequence $U\_n=2^{-n}U$ is a string) you can define a vector space topology by specifying a directed family of strings. If you have an inductive spectrum $(E\_\alpha,i\_\alpha)\_{\alpha\in I}$ where $i\_\alpha:E\_\alpha\to E$ are linear mappings such the union of their ranges span $E$ you consider the family of all strings in $E$ all whose preimages are strings for the given topologies. The basics of such inductive limits are then rather similar to the locally convex case.


If the index set is countable and all $E\_\alpha$ are locally convex the inductive limits in the categories of topological vector spaces and of locally convex spaces coincide -- but this seems to be the only case where you have "regularity results" on the behaviour of bounded or compact sets of the inductive limit like the one of Dieudonne-Schwartz or (if house advertising is permitted) in my Springer Lecture Notes *Derived Functors in Functional Analysis*.


