Topologically independent sets in topological groups and vector spaces

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Topologically independent sets in topological groups and vector spaces

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Topologically independent sets in topological groups and vector spaces

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA22-32829S" target="_blank" >GA22-32829S: Struktura volných Banachových prostorů a jejich druhých duálů</a><br>

Návaznosti

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

Rok uplatnění

2025

Kód důvěrnosti údajů

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

Název periodika

Topology and Its Applications

ISSN

0166-8641

e-ISSN

1879-3207

Svazek periodika

373

Číslo periodika v rámci svazku

11

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

14

Strana od-do

—

Kód UT WoS článku

001583470200001

EID výsledku v databázi Scopus

2-s2.0-105011746798