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

W. Żelazko¹

¹ Institute of Mathematics Polish Academy of Sciences /Sniadeckich 8 00-950 Warszawa, Poland

Studia Mathematica, Tome 150 (2002) no. 3, pp. 295-303

Résumé

Let $X$ be a locally convex space and $L(X)$ be the algebra of all continuous endomorphisms of $X$. It is known (Esterle [2], [3]) 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.

Détail

DOI : 10.4064/sm150-3-6

Keywords: locally convex space algebra continuous endomorphisms known esterle topologizable topological algebra space subnormed sequentially complete condition sufficient obtain other conditions equivalent topologizability exhibit class subnormed spaces called sub banach spaces which necessarily sequentially complete which algebra normable finally exhibit example subnormed space which algebra topologizable