A well-known result for bounded sets in inductive limits of locally convex spaces is the following: If each of the constituent spaces En are Fréchet spaces and E is the inductive limit of the spaces En, then each bounded subset of E is bounded in some En iff E is locally complete. Using DeWilde's localization theorem, we show here that the completeness of each En and the local completeness of E may be replaced with the conditions that the spaces En are all webbed K-spaces and E is locally Baire, respectively.
@article{urn:eudml:doc:42437,
title = {Regular inductive limits of K-spaces.},
journal = {Collectanea Mathematica},
volume = {42},
year = {1991},
pages = {45-49},
zbl = {0772.46001},
mrnumber = {MR1181061},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:42437}
}
Gilsdorf, Thomas E. Regular inductive limits of K-spaces.. Collectanea Mathematica, Tome 42 (1991) pp. 45-49. http://gdmltest.u-ga.fr/item/urn:eudml:doc:42437/