Product of inductive limit topologies on $C_c(X)\times C_c(X)$

Question

I have a stupid question about a topology on $C_c(X)$. Here $X$ is locally compact Hausdorff. Can assume $\sigma$-compact if it helps.

Definition (topology on $C_c(X)$): For each compact $K \subset X$, $C_K(X)$ is the set of functions in $C_c(X)$ with support in $K$. $C_K(X)$ is given a Banach space structure with the sup norm. Call this topology $\tau_K$.

Let $\beta$ be the set of $V \subset C_c(X)$ which are convex, balanced, and which have $V\cap C_K(X) \in \tau_K$ for each compact $K$.

Define $\tau$ to be the collection of sets in $C_c(X)$ given by $\bigcup (\phi_i + W_i)$ where $W_i \in \beta$. This gives a topology on $C_c(X)$ making it a locally convex topological vector space with $\beta$ as a local basis. Moreover, the topology $\tau_K$ coincides with the subspace topology $\tau|_{C_K(X)}$.

For all the above facts see Rudin's functional analysis $\S 6.3$ onwards.

Question: Take the product topology $\tau \times \tau$ on $C_c(X) \times C_c(X)$. Say $V\subset C_c(X)\times C_c(X)$ is convex and balanced. Suppose for every pair of compact sets $K, F \subset X$, we have that $V \cap C_K(X)\times C_F(X) \in \tau_K \times \tau_F$. Is $V$ open in the product topology $\tau\times \tau$?

I came across this question while studying pseudomeasures and induced representations from Folland's Harmonic analysis.

Attempt: Say $(\phi_1, \phi_2) \in V$. It suffices to find $W_1, W_2 \in \beta$ such that $(\phi_1,\phi_2) + W_1\times W_2 \subset V$...Pretty lost at this point.

Also posted here.

Answer 1

The topology $\tau$ you describe makes $(C_c(X),\tau)$ the colimit (or inductive or direct limit) in the category LCS of locally convex spaces of the system $C_K(X)$ with inclusions $i_{K,L}:C_K(X) \hookrightarrow C_L(X)$ for compact subsets $K\subseteq L$ of $X$, i.e., the inclusions $i_K:C_K(X)\to C_c(X)$ have the universal property that, for every family of continuous linear maps (the morphisms of the category) $f_K:C_K(X)\to Y$ with $f_L\circ i_{K,L}=f_K$, there is a unique morphism $f:C_c(X)\to Y$ with $f_K=f\circ i_K$. A big advantage of the categorical viewpoint is that you easily get permanence properties like colimits commute with colimits.

Since the category LCS is additive (you can add two morphisms $(f+g)(x)=f(x)+g(x)$ and this distributes over the composition -- sometimes the term pre-additive category is used) general abstract nonsense tells you that the product is also a coproduct and hence a colimit which commutes with colimits.

Therefore and without any calculations, the product topology $\tau\times \tau$ is the colimit topology of the system $C_K(X)\times C_L(X)$.

EDIT. I hope that the following helps. Whenever you have colimits $E=$colim$_I E_i$ and $F=$colim$_J F_j$ in a category having codropucts (denoted by $\oplus$) it is always true that $E\oplus F =$colim$_{I\times J} E_i\oplus F_j$. Since LCS is additive, coproducts and products are "the same" so that you can replace $\oplus$ by $\times$. (You see that I am quite reluctant to replace the abstract argument by concrete calculations.)

Answer 2

Ok, I think I understand Jochen's wonderful answer. To see if it is so, I try to write some details of what he says.

Category theory (boo!): Define a local base $\alpha$ on $C_c(X) \times C_c(X)$ by the property $V\in \alpha$ if and only if $V$ is convex, balanced, and $V$ intersects every $C_K(X)\times C_L(X)$ openly. One can show that this indeed gives a topology $\omega$ on $C_c(X)\times C_c(X)$ making it a TVS (same argument as in Rudin).

Using the family of continuous maps $C_K(X) \to (C_c(X)\times C_c(X),\omega); f\mapsto(f,0)$, we get a continuous map $\iota_1:(C_c(X),\tau) \to (C_c(X)\times C_c(X), \omega)$. Similarly, we also get a continuous map $\iota_2(g):= (0,g)$.

This gives a continuous map (the identity) $$\iota:=(i_1,i_2):(C_c(X)\times C_c(X), \tau \times \tau) \to (C_c(X)\times C_c(X), \omega).$$ Thus we're done?

Element wise (yay!): Let $\pi_1$ and $\pi_2$ denote the projections from $C_c(X)\times C_c(X)$ to each of the factors. If $V$ is as in the question, we see that $\pi_1\left(\frac{1}{2}V \cap \left(C_K(X)\times \{0\}\right)\right) \subset C_K(X)$ is open. This shows that $V_1:=\pi_1\left(\frac{1}{2}V \cap \left(C_c(X)\times \{0\}\right)\right)$ is open. That is, it belongs to $\tau$. Similarly $V_2:=\pi_2\left(\frac{1}{2}V \cap \left(C_c(X)\times \{0\}\right)\right)$ is in $\tau$. We have that $V_1\times V_2 = V_1\times\{0\}+ \{0\}\times V_2 \subset V$. So we're done?

Not sure if I'm misunderstanding again...