�ber denc-Dual eines topologischen Vektorraumes

Binz, Ernst; Butzmann, Heinz-Peter; Kutzler, Kurt

doi:10.1007/bf01110105

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Lemma 35 Let G Be a Topological Group The Following Assertions Hold1

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2001

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Cited by 13 publications

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“…In [8] it is proved that a locally convex vector space is complete if and only if it is BB-reflexive as a vector space, thus Remark. For any topological vector space E, L c E is BB-reflexive, without any conditions on E [4]. Taking into account that L c E is bicontinuously isomorphic to Γ c E (for any convergence vector space E, Satz 1 of [9]) it can be easily proved that also Γ c E is BBreflexive as a group.…”

Section: Lemma 35 Let G Be a Topological Group The Following Assertions Holdmentioning

confidence: 99%

Completeness properties of locally quasi-convex groups

Bruguera

Chasco

Martín-Peinador

et al. 2001

Topology and its Applications

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It is natural to extend the Grothendieck theorem on completeness, valid for locally convex topological vector spaces, to Abelian topological groups. The adequate framework to do it seems to be the class of locally quasi-convex groups. However, in this paper we present examples of metrizable locally quasi-convex groups for which the analogue to the Grothendieck theorem does not hold. By means of the continuous convergence structure on the dual of a topological group, we also state some weaker forms of the Grothendieck theorem valid for the class of locally quasi-convex groups. Finally, we prove that for the smaller class of nuclear groups, BB-reflexivity is equivalent to completeness.

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Section: Lemma 35 Let G Be a Topological Group The Following Assertions Holdmentioning

confidence: 99%