Characterizations of a Banach Space through the Strong Lacunary and the Lacunary Statistical Summabilities
In this manuscript we characterize the completeness of a normed space through the strong lacunary (<inline-formula> <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>θ</mi> </msub> </semantics> </math> </inline-fo...
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MDPI AG
2020-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/7/1066 |
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doaj-5dca6b642ecf45d5a4894f89cecafaaa2020-11-25T03:18:04ZengMDPI AGMathematics2227-73902020-07-0181066106610.3390/math8071066Characterizations of a Banach Space through the Strong Lacunary and the Lacunary Statistical SummabilitiesSoledad Moreno-Pulido0Giuseppina Barbieri1Fernando León-Saavedra2Francisco Javier Pérez-Fernández3Antonio Sala-Pérez4Department of Mathematics, College of Engineering, University of Cadiz, 11510 Puerto Real, SpainDepartment of Mathematics, University of Salerno, via Giovanni Paolo II, 84084 Fisciano (SA), ItalyDepartment of Mathematics, Faculty of Social Sciences and Communication, University of Cádiz, 11403 Jerez de la Frontera, SpainDepartment of Mathematics, Faculty of Sciences, University of Cádiz, 11510 Puerto Real, SpainDepartment of Mathematics, College of Engineering, University of Cadiz, 11510 Puerto Real, SpainIn this manuscript we characterize the completeness of a normed space through the strong lacunary (<inline-formula> <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>θ</mi> </msub> </semantics> </math> </inline-formula>) and lacunary statistical convergence (<inline-formula> <math display="inline"> <semantics> <msub> <mi>S</mi> <mi>θ</mi> </msub> </semantics> </math> </inline-formula>) of series. A new characterization of weakly unconditionally Cauchy series through <inline-formula> <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>θ</mi> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>S</mi> <mi>θ</mi> </msub> </semantics> </math> </inline-formula> is obtained. We also relate the summability spaces associated with these summabilities with the strong <i>p</i>-Cesàro convergence summability space.https://www.mdpi.com/2227-7390/8/7/1066lacunary statistical summabilitystrong lacunary summabilityweak unconditionally Cauchy series |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Soledad Moreno-Pulido Giuseppina Barbieri Fernando León-Saavedra Francisco Javier Pérez-Fernández Antonio Sala-Pérez |
| spellingShingle |
Soledad Moreno-Pulido Giuseppina Barbieri Fernando León-Saavedra Francisco Javier Pérez-Fernández Antonio Sala-Pérez Characterizations of a Banach Space through the Strong Lacunary and the Lacunary Statistical Summabilities Mathematics lacunary statistical summability strong lacunary summability weak unconditionally Cauchy series |
| author_facet |
Soledad Moreno-Pulido Giuseppina Barbieri Fernando León-Saavedra Francisco Javier Pérez-Fernández Antonio Sala-Pérez |
| author_sort |
Soledad Moreno-Pulido |
| title |
Characterizations of a Banach Space through the Strong Lacunary and the Lacunary Statistical Summabilities |
| title_short |
Characterizations of a Banach Space through the Strong Lacunary and the Lacunary Statistical Summabilities |
| title_full |
Characterizations of a Banach Space through the Strong Lacunary and the Lacunary Statistical Summabilities |
| title_fullStr |
Characterizations of a Banach Space through the Strong Lacunary and the Lacunary Statistical Summabilities |
| title_full_unstemmed |
Characterizations of a Banach Space through the Strong Lacunary and the Lacunary Statistical Summabilities |
| title_sort |
characterizations of a banach space through the strong lacunary and the lacunary statistical summabilities |
| publisher |
MDPI AG |
| series |
Mathematics |
| issn |
2227-7390 |
| publishDate |
2020-07-01 |
| description |
In this manuscript we characterize the completeness of a normed space through the strong lacunary (<inline-formula> <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>θ</mi> </msub> </semantics> </math> </inline-formula>) and lacunary statistical convergence (<inline-formula> <math display="inline"> <semantics> <msub> <mi>S</mi> <mi>θ</mi> </msub> </semantics> </math> </inline-formula>) of series. A new characterization of weakly unconditionally Cauchy series through <inline-formula> <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>θ</mi> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>S</mi> <mi>θ</mi> </msub> </semantics> </math> </inline-formula> is obtained. We also relate the summability spaces associated with these summabilities with the strong <i>p</i>-Cesàro convergence summability space. |
| topic |
lacunary statistical summability strong lacunary summability weak unconditionally Cauchy series |
| url |
https://www.mdpi.com/2227-7390/8/7/1066 |
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