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Some aspects of nuclear vector groups

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Außenhofer L.

Studia Mathematica

2001

tom 146

nr 2

99-113

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).

On the Glicksberg theorem for locally quasi-convex Schwartz groups

100%

Außenhofer L.

Fundamenta Mathematicae

2008

tom 201

nr 2

163-177

We prove that every locally quasi-convex Schwartz group satisfies the Glicksberg theorem for weakly compact sets.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Studia Mathematica

2 Außenhofer L.

1 2008

1 2001

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Some aspects of nuclear vector groups

On the Glicksberg theorem for locally quasi-convex Schwartz groups