Example of an algebra which is nontopologizable as a locally convex topological algebra

Abstract:

Let $X$ be a real or complex linear space and denote by $L(X)$ the algebra of all its endomorphisms. We prove that $L(X)$ is topologizable as a locally convex topological algebra (with jointly continuous multiplication) if and only if it is topologizable as a Banach algebra and this holds if and only if $X$ is of finite dimension.