Answering a question of Halbeisen we prove (by two different methods) that the algebraic dimension of each infinite-dimensional complete linear metric space X equals the size of X. A topological method gives a bit more: the algebraic dimension of a linear metric space X equals |X| provided the hyperspace K(X) of compact subsets of X is a Baire space. Studying the interplay between Baire properties of a linear metric space X and its hyperspace, we construct a hereditarily Baire linear metric space X with meager hyperspace K(X). Also under (d = c) we construct a metrizable separable noncomplete linear metric space with hereditarily Baire hyperspace. We do not know if such a space can be constructed in ZFC.
@article{urn:eudml:doc:41640,
title = {The algebraic dimension of linear metric spaces and Baire properties of their hyperspaces.},
journal = {RACSAM},
volume = {100},
year = {2006},
pages = {31-37},
mrnumber = {MR2267398},
zbl = {1114.46002},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41640}
}
Banakh, Taras; Plichko, Anatolij. The algebraic dimension of linear metric spaces and Baire properties of their hyperspaces.. RACSAM, Tome 100 (2006) pp. 31-37. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41640/