The closure of cones in the algebra of test functions

We prove that the topological closure of the cone K of positive elements in the algebra of test functions is the same for several topologies (K?τo° = K?τ∞). Further we show that all elements of the closure of K are of the form

$\text{∑}\text{I=1}\text{∞}\text{f}{}^{\mathrm{(i)1}}\text{f}{}^{\mathrm{(i)}}\text{with f}{}^{\mathrm{<mtext>(</mtext><mtext>i</mtext><mtext>)</mtext>}}\text{?}\text{L}{}_{\mathrm{?}}$

and the sum converges. Applying these results, we give a characterization of the elements of K?^τ_o too.