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

W. Żelazko. "When is L(X) topologizable as a topological algebra?." Studia Mathematica 150.3 (2002): 295-303. <http://eudml.org/doc/285184>.

@article{W2002,
abstract = {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.},
author = {W. Żelazko},
journal = {Studia Mathematica},
keywords = {topological algebra; algebra of all continuous endomorphisms; composition; sub-Banach spaces},
language = {eng},
number = {3},
pages = {295-303},
title = {When is L(X) topologizable as a topological algebra?},
url = {http://eudml.org/doc/285184},
volume = {150},
year = {2002},
}

TY - JOUR
AU - W. Żelazko
TI - When is L(X) topologizable as a topological algebra?
JO - Studia Mathematica
PY - 2002
VL - 150
IS - 3
SP - 295
EP - 303
AB - 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.
LA - eng
KW - topological algebra; algebra of all continuous endomorphisms; composition; sub-Banach spaces
UR - http://eudml.org/doc/285184
ER -