In [2] W. Banaszczyk introduced nuclear groups, a Hausdorff variety of abelian topological groups which is generated by all nuclear vector groups (cf. 2.3) and which contains all nuclear vector spaces and all locally compact abelian groups. We prove in 5.6 that the Hausdorff variety generated by all nuclear vector spaces and all locally compact abelian groups (denoted by 𝒱₁) is strictly smaller than the Hausdorff variety of all nuclear groups (denoted by 𝒱₂). More precisely, we characterize those nuclear vector groups belonging to 𝒱₁ (5.5). (These are called special nuclear vector groups.) It is proved that special nuclear vector groups can be embedded into a product of nuclear and of discrete vector spaces (2.5). The sequence space Σ₀ is introduced (2.6) and it is proved that it is a nuclear but not a special nuclear vector group (2.12). Moreover, together with all discrete vector spaces it generates the Hausdorff variety of all nuclear groups (3.3). We show that the Hausdorff variety 𝒱₀ generated by all nuclear vector spaces is strictly contained in 𝒱₁ (4.5).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm146-2-1,
author = {Lydia Au\ss enhofer},
title = {Some aspects of nuclear vector groups},
journal = {Studia Mathematica},
volume = {147},
year = {2001},
pages = {99-113},
zbl = {0976.22001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm146-2-1}
}
Lydia Außenhofer. Some aspects of nuclear vector groups. Studia Mathematica, Tome 147 (2001) pp. 99-113. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm146-2-1/