details to be checked…
Let be a locally compact Hausdorff space. The inductive limit topology on is with respect to the family , where runs over all compact subsets of and is the inclusion map of in .
Proof. As the supremum norm topology on is a convex topology such that each is continuous, the inductive limit topology is finer than supremum norm topology on , and thus the restriction of inductive limit topology on is finer than the supremum norm topology on .
For any open set in the inductive limit topology on , is an open set in the supremum topology on , i.e., the restriction of inductive limit topology on is coarser than the supremum norm topology on . ◻
Reference: Bourbaki, Topological Vector Spaces, II.29