Topologically independent sets in topological groups and vector spaces

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21240%2F25%3A00384777" target="_blank" >RIV/68407700:21240/25:00384777 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1016/j.topol.2025.109527" target="_blank" >https://doi.org/10.1016/j.topol.2025.109527</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.topol.2025.109527" target="_blank" >10.1016/j.topol.2025.109527</a>

Result language

angličtina

Original language name

Topologically independent sets in topological groups and vector spaces

Original language description

We study topological versions of an independent set in an abelian group and a linearly independent set in a vector space, a topologically independent set in a topological group and a topologically linearly independent set in a topological vector space. These counterparts of their algebraic versions are defined analogously and possess similar properties. Let Cx be the multiplicative group of the field of complex numbers with its usual topology. We prove that a subset A of an arbitrary Tychonoff power of is topologically independent if and only if the topological subgroup that it generates is the Tychonoff direct sum . This theorem substantially generalizes an earlier result of the author, who has proved this for Abelian precompact groups. Further, we show that topologically independent and topologically linearly independent sets coincide in vector spaces with weak topologies, although they are different in general. We characterize topologically linearly independent sets in vector spaces with weak topologies and normed spaces. In a weak topology, a set A is topologically linearly independent if and only if its linear span is the Tychonoff direct sum . In normed spaces A is topologically linearly independent if and only if it is uniformly minimal. Thus, from the point of view of topological linear independence, the Tychonoff direct sums and (linear spans of) uniformly minimal sets, which are closely related to bounded biorthogonal systems, are of the same essence. We also provide an application of topological linear independence to Lipschitz-free spaces.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GA22-32829S" target="_blank" >GA22-32829S: The structure of free Banach spaces and of their second duals</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2025

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Topology and Its Applications

ISSN

0166-8641

e-ISSN

1879-3207

Volume of the periodical

373

Issue of the periodical within the volume

11

Country of publishing house

NL - THE KINGDOM OF THE NETHERLANDS

Number of pages

14

Pages from-to

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UT code for WoS article

001583470200001

EID of the result in the Scopus database

2-s2.0-105011746798