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

Tom 150 / 2002

W. Żelazko Studia Mathematica 150 (2002), 295-303 MSC: 47L10, 46H35. DOI: 10.4064/sm150-3-6


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.


  • W. ŻelazkoInstitute of Mathematics
    Polish Academy of Sciences
    /Sniadeckich 8
    00-950 Warszawa, Poland

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