Direct sums and products in topological groups and vector spaces

Result code in 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>

Result on the web

<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>

Result language

angličtina

Original language name

Direct sums and products in topological groups and vector spaces

Original language description

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.

Czech name

—

Czech description

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Type

J_x - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)

CEP classification

BA - General mathematics

OECD FORD branch

—

Project

<a href="/en/project/GPP201%2F12%2FP724" target="_blank" >GPP201/12/P724: Relations between topological spaces and their topological groups of G-valued continuous functions for a given topological group G</a><br>

Continuities

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

Publication year

2016

Confidentiality

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

Name of the periodical

Journal of Mathematical Analysis and Applications

ISSN

0022-247X

e-ISSN

—

Volume of the periodical

2016

Issue of the periodical within the volume

437

Country of publishing house

US - UNITED STATES

Number of pages

26

Pages from-to

1257-1282

UT code for WoS article

000370312500028

EID of the result in the Scopus database

2-s2.0-84957434564