Some aspects of nuclear vector groups

Lydia Außenhofer. "Some aspects of nuclear vector groups." Studia Mathematica 146.2 (2001): 99-113. <http://eudml.org/doc/284651>.

@article{LydiaAußenhofer2001,
abstract = { 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). },
author = {Lydia Außenhofer},
journal = {Studia Mathematica},
keywords = {nuclear group; nuclear locally convex space; nuclear vector group; variety of groups; locally compact groups; nuclear vector spaces},
language = {eng},
number = {2},
pages = {99-113},
title = {Some aspects of nuclear vector groups},
url = {http://eudml.org/doc/284651},
volume = {146},
year = {2001},
}

TY - JOUR
AU - Lydia Außenhofer
TI - Some aspects of nuclear vector groups
JO - Studia Mathematica
PY - 2001
VL - 146
IS - 2
SP - 99
EP - 113
AB - 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).
LA - eng
KW - nuclear group; nuclear locally convex space; nuclear vector group; variety of groups; locally compact groups; nuclear vector spaces
UR - http://eudml.org/doc/284651
ER -