Some aspects of nuclear vector groups

Lydia Außenhofer¹

¹ Mathematisches Institut Auf der Morgenstelle 10 D-72076 Tübingen, Germany

Studia Mathematica, Tome 146 (2001) no. 2, pp. 99-113

Résumé

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 ${\cal V}_1$) is strictly smaller than the Hausdorff variety of all nuclear groups (denoted by ${\cal V}_2$). More precisely, we characterize those nuclear vector groups belonging to ${\cal V}_1$ (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 ${\mit \Sigma }_0$ 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 ${\cal V}_0$ generated by all nuclear vector spaces is strictly contained in ${\cal V}_1$ (4.5).

Détail

DOI : 10.4064/sm146-2-1

Keywords: banaszczyk introduced nuclear groups hausdorff variety abelian topological groups which generated nuclear vector groups which contains nuclear vector spaces locally compact abelian groups prove hausdorff variety generated nuclear vector spaces locally compact abelian groups denoted cal strictly smaller hausdorff variety nuclear groups denoted cal precisely characterize those nuclear vector groups belonging cal these called special nuclear vector groups proved special nuclear vector groups embedded product nuclear discrete vector spaces sequence space mit sigma introduced proved nuclear special nuclear vector group moreover together discrete vector spaces generates hausdorff variety nuclear groups hausdorff variety cal generated nuclear vector spaces strictly contained cal