Show that if two distributions have the same kernel, then one is a constant multiple of the other.

Question

If the set of test function corresponding to the kernel of distribution f coincides with the kernel of distribution g, then the distribution f is a constant multiple of g.

Answer 1

Set $K =\ker f = \ker g$. Quotienting, we obtain isomorphisms of vector spaces $\tilde f,\tilde g :\mathcal D/K \to \mathbb R$, which means that $\tilde f \tilde g^{-1}$ is an isomorphism between vector spaces $\mathbb R\to\mathbb R$. The only such maps are multiplication by constants. To finish, note that $f = \tilde f \tilde g^{-1} g$.