In what follows "X" denotes a locally compact topological space. The continuous real-valued functions with compact support on "X" form a vector space formula_1, which can be given a natural locally convex topology. Indeed, formula_2 is the union of the spaces formula_3 of continuous functions with support contained in compact sets "K". Each of the spaces formula_3 carries naturally the topology of uniform convergence, which makes it into a Banach space. But as a union of topological spaces is a special case of a direct limit of topological spaces, the space formula_2 can be equipped with the direct limit locally convex topology induced by the spaces formula_3; this topology is finer than the topology of uniform convergence.