# Direct (Inductive) limit of (locally convex) TVEs and universal property

This is not really a question, I'd just like to discuss a little about universal properties (more specifically, the direct limit) in TVEs.

I'm trying to work with universal properties in Topological Vector Spaces (TVEs) (over $\mathbb{K}\in\left\{\mathbb{R},\mathbb{C}\right\}$), constructing the universal objects, etc... If we were to define an inductive limit of TVEs, according to the categorical definition, it'ld be the following:

Fix a directed set $(I,\leq)$, that is, an ordered set $(I,\leq)$ such that for every $i,j\in I$ there exists $k\in I$ such that $i\leq k$ and $j\leq k$ (although it's common to drop antissimmetry).

Definition: Let $(I,\leq)$ be a directed set and $\left\{V_i:i\in I\right\}$ be a family of TVEs and $\left\{T_{ji}:i\leq j\in I\right\}$ be a family of functions $T_{ji}\in\mathscr{L}(V_i,V_j)$, that is, continuous linear functions from $V_i$ to $V_j$ satisfying:

1. $T_{ii}=id_{V_i}$
2. $T_{ki}=T_{kj}T_{ji}$, for all $i\leq j\leq k$.

A direct limit of those two families is a TVE $\mathbb{V}$ together with a family $\left\{\phi_i\in\mathscr{L}(V_i,\mathbb{V}):i\in I\right\}$ satisfying $\phi_i=\phi_jT_{ji}$ for $i\leq j$ and such that for every other TVE $\mathbb{W}$ with a family of morphisms $\left\{\psi_i\in\mathscr{L}(V_i,\mathbb{W}):i\in I\right\}$ satisfying $\psi_i=\psi_jT_{ji}$ for $j\leq i$, there exists an unique function $\Phi\in\mathscr{L}(\mathbb{V},\mathbb{W})$ satisfying $\psi_i=\Phi\phi_i$.

The same definition works for locally convex TVEs. Although, most books define the direct limit of locally convex spaces in the following way:

Definition (H. H. Schaefer, Topological Vector Spaces): Let $\left\{V_\lambda:\lambda\in\Lambda\right\}$ be a family of locally convex spaces (where $\Lambda$ is only an index set), $\mathbb{V}$ a vector space and $\left\{T_\lambda\in L(V_\lambda,\mathbb{V}):\lambda\in\Lambda\right\}$. The direct limit topology on $\mathbb{V}$ is the finest locally convex topology for which each $T_\lambda$ is continuous.

It's not obvious to me that those two definition are equivalent. I think that the first definition is more ellegant than the second. Is there a good reason why we should use the second one?

One thing that is funny is that the direct sum of locally convex TVEs is defined as the algebraic direct sum (the subspace of $\prod_{\lambda\in\Lambda}V_\lambda$ of elements $(v_\lambda)$ for which $v_\lambda=0$ except for a finite number of indexes) with the direct limit topology, and it satisfies the universal property of the direct sum. When working in categories, this is usually done the other way: the direct limit is a quotient of the direct sum (if that makes sense, of course).

• The reason is mainly historical: direct limits of topological vector spaces appeared first in the work of Dieudonné-Schwartz. The space $\mathbb{V}$ was fixed (test functions) and what they were looking for was a convenient topology. The categorical formulation of colimits came much later. // Your last point is an instance of a general fact: in every category there is a canonical map from the coproduct into the product (if they exist). Here it happens to be injective. The product is very familiar, so it is convenient to describe the underlying set of the coproduct as a subset of the product. Commented May 6, 2013 at 1:59

They are, of course, not equivalent. Your second definition is for what is called $final$ locally convex topology on $\mathbb V$ w.r.t. $T_\lambda$. I believe Bourbaki's TVS book has a good account on that.
On your first definition, locally convex direct limit of the directed system $V=(V_i,T_{ij})_{i,j\in I}$ is the algebraic direct limit $(\mathbb{V},\phi_i)_{i\in I}$ of the system $V$ as of the system of vector spaces, equipped with the final locally convex topology w.r.t. $\phi_i:V_i\to\mathbb{V}$.