 The test functions are themselves locally integrable, and so define distributions.As such they are dense in D′("U") with respect to the topology on D′("U") in the sense that for any distribution "T" ∈ D′("U"), there is a net "formula_1""i" ∈ D("U") such that

for all "Ψ" ∈ D("U").This fact follows from the Hahn–Banach theorem, since the dual of D′("U") with its weak-* topology is the space D("U").