# Colimits in the category of (not necessarily locally convex) topological vector spaces

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?)

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.

I found a construction of the coproduct in the category of topological abelian groups from this reference: https://core.ac.uk/download/pdf/82771298.pdf, which can be also applied to the category of topological vector spaces without any difficulty.

The construction goes as follows.

1. First, form the algebraic direct sum $$E:=\bigoplus_{i\in I}E_{i}$$ of tvs's $$E_{i}$$'s. Consider the collection $$\mathcal{P}$$ of pairs $$(N,\mathcal{T})$$, where $$N$$ is a subspace of $$E$$ and $$\mathcal{T}$$ is a vector space topology on $$E/N$$ making the composition $$E_{i}\rightarrow E\rightarrow(E/N,\mathcal{T})$$ continuous.

2. Embed $$E$$ into the product space $$\prod_{(N,\mathcal{T})\in\mathcal{P}}(E/N,\mathcal{T})$$. Since the product space contains a copy of each $$E_{i}$$, this embedding is injective.

3. Then it is routine to check that $$E$$ endowed with the subspace topology is the coproduct.

As noted in comments, hence arbitrary colimits can be formed by passing to a quotient. However, the same construction perhaps can be still applied to arbitrary filtered colimits, though as noted in the reference this construction is only useful for showing existence and not really usable in practice.

In the note http://www-users.math.umn.edu/~garrett/m/fun/uncountable_coproducts.pdf it is proven that an uncountable (not-necessarily locally convex) coproduct of lines does not exist, by using the spaces $$\ell^p$$ with $$0, which are not locally convex.

EDIT/Correction: oops, yes, sorry, as in comments, what is proven there is not that no possible coproduct exists in the larger category, but that the locally convex coproduct is not a coproduct in the larger category.

Further, the device of using the non-locally-convex $$\ell^p$$ spaces does not seem to immediately give non-existence of a coproduct in the larger category (though I may be mistaken).

• This is brought from that note: "Without directly considering the not-necessarily-locally-convex coproduct, we can show that the locally convex coproduct of an uncountable number of locally convex topological vector spaces fails to be a coproduct in the larger category of not-necessarily-locally-convex topological vector spaces. " But I failed to understand why it shows that the uncountable coproduct of C does not exist in the category of TVS's. Nov 1, 2019 at 17:03
• For clarification, this is what I understood: It certainly shows that the coproduct of C's in the category of locally convex spaces is NOT the coproduct of C's in the TVS's. But that does not show that the coproduct in the TVS's cannot exist. Nov 1, 2019 at 17:12
• @user131781 Actually, I'm curious of how to form the direct sum. If there are finitely many summands, it should coincide with the product according to my understanding, but is it still true for infinite summands case? Nov 1, 2019 at 20:38
• I had meant them for the second question. At the moment I have no access to mathscinet or a good mathematical library so I am afraid I can’t be of any more help. Nov 2, 2019 at 3:55
• @JunekeyJeon My suggestion is to check exercise 1 (b) of Schaefer's first chapter, and exercise 12 of chapter II. I have not yet had the time to do these myself, though. Nov 3, 2019 at 1:21