A note on multiplier convergent series

Result code in 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>

Result on the web

<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>

Result language

angličtina

Original language name

A note on multiplier convergent series

Original language description

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 $Z^{(N)}$).

Czech name

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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

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Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2016

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

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Volume of the periodical

2016

Issue of the periodical within the volume

211

Country of publishing house

NL - THE KINGDOM OF THE NETHERLANDS

Number of pages

10

Pages from-to

28-37

UT code for WoS article

000384781000004

EID of the result in the Scopus database

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