A solution of the one dimensional Vlasov equation is compared to the motion of a system of particles interacting through the one dimensional Coulomb potential. For each value of the time t, both the distribution function f(t) and the system of point particles II (t) are considered as elements of the dual space of a suitable vector space of test functions. It is shown that for a fixed solution f(t) of the Vlasov equation, if II (0) tends to f(o) in the chosen dual space, then n(t) tends to f(t) for all t > 0.