Let X be a locally convex space and L(X) be the algebra of all continuous endomorphisms of X. It is known (Esterle , ) that if L(X) is topologizable as a topological algebra, then the space X is subnormed. We show that in the case when X is sequentially complete this condition is also sufficient. In this case we also obtain some other conditions equivalent to the topologizability of L(X). We also exhibit a class of subnormed spaces X, called sub-Banach spaces, which are not necessarily sequentially complete, but for which the algebra L(X) is normable. Finally we exhibit an example of a subnormed space X for which the algebra L(X) is not topologizable.