Parts of Banach algebra theory have been generalized recently [2; 1 ] to multiplicatively convex topological algebras. An important class of examples of Banach algebras is the class of subalgebras of the algebra of continuous linear transformations on a Banach space. It is natural to ask whether an algebra of continuous linear transformations on a convex vector space, with the topology of bounded convergence (the analogue of the norm topology in an algebra of transformations on a Banach space), is a multiplicatively convex topological algebra. We show that this is in general false, in fact that if the algebra is large enough multiplication is not even continuous unless the convex vector space is normable.