赋范空间的离散子群是自由的

IF 0.9 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-03-17 DOI:10.1112/blms.70051

Tomasz Kania, Ziemowit Kostana

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引用次数: 0

摘要

Ancel， Dobrowolski和Grabowski （Studia Math. 109(1994): 277-290）证明了有模空间的可加群的每一个可数离散子群都是自由阿别的，因此与整数的可加群的若干副本的直和同态。本文采用基于初等子模型理论和奇异紧性定理的集合论方法，从它们的结果中去掉了基数约束，证明了赋范空间的加性群的每一个离散子群都是自由阿贝尔的。

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Discrete subgroups of normed spaces are free

Ancel, Dobrowolski and Grabowski (Studia Math. 109 (1994): 277–290) proved that every countable discrete subgroup of the additive group of a normed space is free Abelian, hence isomorphic to the direct sum of a certain number of copies of the additive group of the integers. In the present paper, we take a set-theoretic approach based on the theory of elementary submodels and the Singular Compactness Theorem to remove the cardinality constraint from their result and prove that indeed every discrete subgroup of the additive group of a normed space is free Abelian.

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