When is $L(X)$ topologizable as a topological algebra?
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.