Direct sums and products in topological groups and vector spaces

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F44555601%3A13440%2F16%3A43887078" target="_blank" >RIV/44555601:13440/16:43887078 - isvavai.cz</a>

Výsledek na webu

<a href="http://www.sciencedirect.com/science/article/pii/S0022247X16000627" target="_blank" >http://www.sciencedirect.com/science/article/pii/S0022247X16000627</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.jmaa.2016.01.037" target="_blank" >10.1016/j.jmaa.2016.01.037</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Direct sums and products in topological groups and vector spaces

Popis výsledku v původním jazyce

We call a subset $A$ of an abelian topological group $G$: (i) {em absolutely Cauchy summable/} provided that for every open neighbourhood $U$ of $0$ one can find a finite set $Fsubseteq A$ such that the subgroup generated by $Asetminus F$ is contained in $U$; (ii) {em absolutely summable/} if, for every family ${z_a:ain A}$ of integer numbers, there exists $gin G$ such that the net $left{sum_{ain F} z_a a: Fsubseteq Ambox{ is finite}right}$ converges to $g$; (iii) {em topologically independent/} provided that $0not in A$ and for every neighbourhood $W$ of $0$ there exists a neighbourhood $V$ of $0$ such that, for every finite set $Fsubseteq A$ and each set ${z_a:ain F}$ of integers, $sum_{ain F}z_aain V$ implies that $z_aain W$ for all $ain F$. We prove that: (1) an abelian topological group contains a direct product (direct sum) of $kappa$-many non-trivial topological groups if and only if it contains a topologically independent, absolutely (Cauchy) summable subset of cardinality $kappa$; (2) a topological vector space contains $R^{(N)}$ as its subspace if and only if it has an infinite absolutely Cauchy summable set; (3) a topological vector space contains $R^{N}$ as its subspace if and only if it has an $R^N$ multiplier convergent series of non-zero elements. We answer a question of Huv{s}ek and generalize results by Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki.

Název v anglickém jazyce

Direct sums and products in topological groups and vector spaces

Popis výsledku anglicky

We call a subset $A$ of an abelian topological group $G$: (i) {em absolutely Cauchy summable/} provided that for every open neighbourhood $U$ of $0$ one can find a finite set $Fsubseteq A$ such that the subgroup generated by $Asetminus F$ is contained in $U$; (ii) {em absolutely summable/} if, for every family ${z_a:ain A}$ of integer numbers, there exists $gin G$ such that the net $left{sum_{ain F} z_a a: Fsubseteq Ambox{ is finite}right}$ converges to $g$; (iii) {em topologically independent/} provided that $0not in A$ and for every neighbourhood $W$ of $0$ there exists a neighbourhood $V$ of $0$ such that, for every finite set $Fsubseteq A$ and each set ${z_a:ain F}$ of integers, $sum_{ain F}z_aain V$ implies that $z_aain W$ for all $ain F$. We prove that: (1) an abelian topological group contains a direct product (direct sum) of $kappa$-many non-trivial topological groups if and only if it contains a topologically independent, absolutely (Cauchy) summable subset of cardinality $kappa$; (2) a topological vector space contains $R^{(N)}$ as its subspace if and only if it has an infinite absolutely Cauchy summable set; (3) a topological vector space contains $R^{N}$ as its subspace if and only if it has an $R^N$ multiplier convergent series of non-zero elements. We answer a question of Huv{s}ek and generalize results by Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki.

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

—

Projekt

<a href="/cs/project/GPP201%2F12%2FP724" target="_blank" >GPP201/12/P724: Závislosti topologických prostorů a jejich topologických grup G-hodnotových spojitých funkcí pro danou topologickou grupu G</a><br>

Návaznosti

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

Rok uplatnění

2016

Kód důvěrnosti údajů

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

Název periodika

Journal of Mathematical Analysis and Applications

ISSN

0022-247X

e-ISSN

—

Svazek periodika

2016

Číslo periodika v rámci svazku

437

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

26

Strana od-do

1257-1282

Kód UT WoS článku

000370312500028

EID výsledku v databázi Scopus

2-s2.0-84957434564