Some aspects of nuclear vector groups

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 V-1) is strictly smaller than the Hausdorff variety of all nuclear groups (denoted by V-2). More precisely, we characterize those nuclear vector groups belonging to 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 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 V-0 generated by all nuclear vector spaces is strictly contained in V-1 (4.5).