Joint continuity of tensor product

Question

Let $X, Y$ be locally compact Hausdorff spaces and consider the spaces $\mathcal{K}_{\mathbb{C}}(X), \mathcal{K}_{\mathbb{C}}(Y)$ of continuous functions with compact support on $X$ and $Y$ respectively. Consider these spaces with their natural inductive limit topologies (the ones determined by the subspaces $\mathcal{K}_{\mathbb{C}}(X, K) := \{f \in\ \mathcal{K}_{\mathbb{C}}(X)| suppf \subset K \}$ , where $K \subset X$ is compact, and these subspaces are endowed with the sup-norm topology; same for $Y$). My question is that is the tensor product $$ \begin{cases} \begin{align*} \otimes:\mathcal{K}_{\mathbb{C}}(X)\times\mathcal{K}_{\mathbb{C}}(Y) &\to \mathcal{K}_{\mathbb{C}}(X \times Y)\\ (f,g) \mapsto f \otimes g &= ((x,y) \mapsto f(x)g(y)) \end{align*} \end{cases} $$ jointly continuous ($\mathcal{K}_{\mathbb{C}}(X \times Y)$ is endowed with the inductive topology too)? I was able to show that it is separately continuous but in general that's not enough. It is also known that the spaces above are barrelled, locally convex and Hausdorff.I would appreciate any suggestions!