A note on multiplier convergent series

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F44555601%3A13440%2F16%3A43887712" target="_blank" >RIV/44555601:13440/16:43887712 - isvavai.cz</a>

Výsledek na webu

<a href="http://www.sciencedirect.com/science/article/pii/S0166864116301870" target="_blank" >http://www.sciencedirect.com/science/article/pii/S0166864116301870</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.topol.2016.08.013" target="_blank" >10.1016/j.topol.2016.08.013</a>

Jazyk výsledku

angličtina

Název v původním jazyce

A note on multiplier convergent series

Popis výsledku v původním jazyce

Given a topological ring $R$ and $mathscr{F}subset R^N$ a (formal) series $sum_{ninN}x_n$ in a topological $R$-module $E$ is {em $mathscr{F}$ multiplier convergent in $E$} (respectively {em $mathscr{F}$ multiplier Cauchy in $E$}) provided that the sequence ${sum_{i=0}^nr(i)x_i:ninN}$ of partial sums converges (respectively, is a Cauchy sequence) for every sequence function $rinmathscr{F}$. In this paper we investigate for which $mathscr{G}subset R^N$ every $mathscr{F}$ multiplier convergent (Cauchy) series is also $mathscr{G}$ multiplier convergent (Cauchy). We obtain some general theorems about the Cauchy version of this problem. In particular, we prove that every $Z^N$ multiplier Cauchy series is already $R^N$ multiplier Cauchy in every topological vector space. On the other hand, we construct examples that in particular show that a $Z^N$ multiplier convergent series need not to be even $Q^N$ multiplier convergent and that there are topological vector spaces containing non-trivial $Q^N$ multiplier convergent series that do not contain non-trivial $R^N$ convergent series. As a consequence of this example, there are topological vector spaces containing the topological group $Q^N$ (and thus $Z^N$ and $Z^{(N)}$ as well) that do not contain the topological vector space $R^N$. On the contrary, it was proved in cite{DSS}, that a sequentially complete topological vector space that contains the topological group $Z^{(N)}$ must already contain the topological vector space $R^N$. Hence our example demonstrates, that in the latter result, the condition of sequential completeness can not be weakened by assuming that the space in question contains the topological group $Z^N$ (which is the sequential completion of $Z^{(N)}$).

Název v anglickém jazyce

A note on multiplier convergent series

Popis výsledku anglicky

Given a topological ring $R$ and $mathscr{F}subset R^N$ a (formal) series $sum_{ninN}x_n$ in a topological $R$-module $E$ is {em $mathscr{F}$ multiplier convergent in $E$} (respectively {em $mathscr{F}$ multiplier Cauchy in $E$}) provided that the sequence ${sum_{i=0}^nr(i)x_i:ninN}$ of partial sums converges (respectively, is a Cauchy sequence) for every sequence function $rinmathscr{F}$. In this paper we investigate for which $mathscr{G}subset R^N$ every $mathscr{F}$ multiplier convergent (Cauchy) series is also $mathscr{G}$ multiplier convergent (Cauchy). We obtain some general theorems about the Cauchy version of this problem. In particular, we prove that every $Z^N$ multiplier Cauchy series is already $R^N$ multiplier Cauchy in every topological vector space. On the other hand, we construct examples that in particular show that a $Z^N$ multiplier convergent series need not to be even $Q^N$ multiplier convergent and that there are topological vector spaces containing non-trivial $Q^N$ multiplier convergent series that do not contain non-trivial $R^N$ convergent series. As a consequence of this example, there are topological vector spaces containing the topological group $Q^N$ (and thus $Z^N$ and $Z^{(N)}$ as well) that do not contain the topological vector space $R^N$. On the contrary, it was proved in cite{DSS}, that a sequentially complete topological vector space that contains the topological group $Z^{(N)}$ must already contain the topological vector space $R^N$. Hence our example demonstrates, that in the latter result, the condition of sequential completeness can not be weakened by assuming that the space in question contains the topological group $Z^N$ (which is the sequential completion of $Z^{(N)}$).

Druh

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

CEP obor

BA - Obecná matematika

OECD FORD obor

—

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2016

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

—

Svazek periodika

2016

Číslo periodika v rámci svazku

211

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

10

Strana od-do

28-37

Kód UT WoS článku

000384781000004

EID výsledku v databázi Scopus

—