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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm150-3-6,
author = {W. \.Zelazko},
title = {When is L(X) topologizable as a topological algebra?},
journal = {Studia Mathematica},
volume = {151},
year = {2002},
pages = {295-303},
zbl = {1006.47052},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm150-3-6}
}
W. Żelazko. When is L(X) topologizable as a topological algebra?. Studia Mathematica, Tome 151 (2002) pp. 295-303. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm150-3-6/