  We give a simple method for constructing commutative Frechet algebras which
admit two inequivalent Frechet algebra topologies. The result is applied to
show that the action of any non-algebraic analytic function may fail to be
uniquely defined among other useful applications. We give an affirmative answer
to a question of Loy from 1974. We also obtain the uniqueness of the Frechet
algebra topology of certain Frechet algebras with finite dimensional radicals.
