Lineability and modes of convergence
In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in measure but not a.e. pointwise, uniformly but not pointwise conv...
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| Published in: | Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Vol. 114; no. 1 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
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Springer International Publishing
2020
Springer Nature B.V |
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| Abstract | In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in measure but not a.e. pointwise, uniformly but not pointwise convergent, and uniformly convergent but not in
L
1
-norm, are analyzed. These findings extend and complement a number of earlier results by several authors. |
|---|---|
| AbstractList | In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in measure but not a.e. pointwise, uniformly but not pointwise convergent, and uniformly convergent but not in L1-norm, are analyzed. These findings extend and complement a number of earlier results by several authors. In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in measure but not a.e. pointwise, uniformly but not pointwise convergent, and uniformly convergent but not in L 1 -norm, are analyzed. These findings extend and complement a number of earlier results by several authors. |
| ArticleNumber | 18 |
| Author | Calderón-Moreno, M. C. Gerlach-Mena, P. J. Prado-Bassas, J. A. |
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| Cites_doi | 10.1007/s13398-016-0318-y 10.1016/j.jfa.2013.11.014 10.1016/j.jmaa.2010.12.061 10.1090/S0002-9939-2012-11466-2 10.1016/j.laa.2010.11.012 10.1016/j.jmaa.2018.10.007 10.1016/j.laa.2015.09.009 10.4064/sm8358-10-2016 10.1016/j.top.2009.11.013 10.1090/S0002-9939-04-07533-1 10.1090/S0273-0979-2013-01421-6 10.1307/mmj/1441116657 |
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| Keywords | Uniform convergence Lineability Secondary 15A03 Convergence in Primary 28A20 40A05 Convergence in measure Pointwise convergence Algebrability norm |
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| References | Kitson, Timoney (CR13) 2011; 378 Botelho, Diniz, Favaro, Pellegrino (CR8) 2011; 434 Bernal-González, Pellegrino, Seoane-Sepúlveda (CR7) 2014; 51 Gurariy (CR12) 1966; 167 Aron, Bernal-González, Pellegrino, Seoane-Sepúlveda (CR2) 2016 Calderón-Moreno, Gerlach-Mena, Prado-Bassas (CR10) 2019; 470 Botelho, Favaro (CR9) 2015; 64 Bernal-González, Ordóñez Cabrera (CR6) 2014; 266 Nogueira, Pellegrino (CR15) 2015; 487 Araújo, Bernal-González, Muñoz-Fernández, Prado-Bassas, Seoane-Sepúlveda (CR1) 2017; 237 Oxtoby (CR16) 1980 Barroso, Botelho, Favaro, Pellegrino (CR5) 2013; 141 Aron, Gurariy, Seoane-Sepúlveda (CR4) 2005; 133 Conejero, Fenoy, Murillo-Arcila, Seoane-Sepúlveda (CR11) 2017; 111 Aron, García-Pacheco, Pérez-García, Seoane-Sepúlveda (CR3) 2009; 48 Nielsen (CR14) 1997 OA Nielsen (743_CR14) 1997 MC Calderón-Moreno (743_CR10) 2019; 470 RM Aron (743_CR2) 2016 VI Gurariy (743_CR12) 1966; 167 CS Barroso (743_CR5) 2013; 141 L Bernal-González (743_CR6) 2014; 266 G Botelho (743_CR8) 2011; 434 RM Aron (743_CR3) 2009; 48 RM Aron (743_CR4) 2005; 133 L Bernal-González (743_CR7) 2014; 51 JA Conejero (743_CR11) 2017; 111 JC Oxtoby (743_CR16) 1980 D Kitson (743_CR13) 2011; 378 G Araújo (743_CR1) 2017; 237 G Botelho (743_CR9) 2015; 64 T Nogueira (743_CR15) 2015; 487 |
| References_xml | – volume: 111 start-page: 673 year: 2017 end-page: 684 ident: CR11 article-title: Lineability within probability theory settings publication-title: Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM doi: 10.1007/s13398-016-0318-y contributor: fullname: Seoane-Sepúlveda – volume: 266 start-page: 3997 year: 2014 end-page: 4025 ident: CR6 article-title: Lineability criteria, with applications publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2013.11.014 contributor: fullname: Ordóñez Cabrera – year: 2016 ident: CR2 publication-title: Lineability: The Search for Linearity in Mathematics, Monographs and Research Notes in Mathematics contributor: fullname: Seoane-Sepúlveda – volume: 378 start-page: 680 year: 2011 end-page: 686 ident: CR13 article-title: Operator ranges and spaceability publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2010.12.061 contributor: fullname: Timoney – year: 1980 ident: CR16 publication-title: Measure and Category, 2nd edn, Graduate Text in Mathematics contributor: fullname: Oxtoby – volume: 141 start-page: 1913 year: 2013 end-page: 1923 ident: CR5 article-title: Lineability and spaceability for the weak form of Peano’s theorem and vector-valued sequence spaces publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-2012-11466-2 contributor: fullname: Pellegrino – volume: 434 start-page: 1255 year: 2011 end-page: 1260 ident: CR8 article-title: Spaceability in Banach and quasi-Banach sequence spaces publication-title: Linear Algebra Appl. doi: 10.1016/j.laa.2010.11.012 contributor: fullname: Pellegrino – volume: 470 start-page: 348 year: 2019 end-page: 359 ident: CR10 article-title: Algebraic structure of continuous, unbounded and integrable functions publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2018.10.007 contributor: fullname: Prado-Bassas – volume: 487 start-page: 172 year: 2015 end-page: 183 ident: CR15 article-title: On the size of certain subsets of invariant Banach sequence spaces publication-title: Linear Algebra Appl. doi: 10.1016/j.laa.2015.09.009 contributor: fullname: Pellegrino – year: 1997 ident: CR14 publication-title: An Introduction to Integration and Measure Theory, Canadian Mathematical Society Series of Monographs and Advanced Texts contributor: fullname: Nielsen – volume: 237 start-page: 119 year: 2017 end-page: 136 ident: CR1 article-title: Lineability in sequence and function spaces publication-title: Stud. Math. doi: 10.4064/sm8358-10-2016 contributor: fullname: Seoane-Sepúlveda – volume: 167 start-page: 971 year: 1966 end-page: 973 ident: CR12 article-title: Subspaces and bases in spaces of continuous functions (Russian) publication-title: Dokl. Akad. Nauk SSSR contributor: fullname: Gurariy – volume: 48 start-page: 149 year: 2009 end-page: 156 ident: CR3 article-title: On dense-lineability of sets of functions on publication-title: Topology doi: 10.1016/j.top.2009.11.013 contributor: fullname: Seoane-Sepúlveda – volume: 133 start-page: 795 year: 2005 end-page: 803 ident: CR4 article-title: Lineability and spaceability of sets of functions on publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-04-07533-1 contributor: fullname: Seoane-Sepúlveda – volume: 51 start-page: 71 year: 2014 end-page: 130 ident: CR7 article-title: Linear subsets of nonlinear sets in topological vector spaces publication-title: Bull. Am. Math. Soc. (N.S.) doi: 10.1090/S0273-0979-2013-01421-6 contributor: fullname: Seoane-Sepúlveda – volume: 64 start-page: 539 year: 2015 end-page: 554 ident: CR9 article-title: Constructing Banach spaces of vector-valued sequences with special properties publication-title: Mich. Math. 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Appl. doi: 10.1016/j.jmaa.2010.12.061 contributor: fullname: D Kitson – volume-title: An Introduction to Integration and Measure Theory, Canadian Mathematical Society Series of Monographs and Advanced Texts year: 1997 ident: 743_CR14 contributor: fullname: OA Nielsen – volume-title: Measure and Category, 2nd edn, Graduate Text in Mathematics year: 1980 ident: 743_CR16 contributor: fullname: JC Oxtoby – volume: 434 start-page: 1255 year: 2011 ident: 743_CR8 publication-title: Linear Algebra Appl. doi: 10.1016/j.laa.2010.11.012 contributor: fullname: G Botelho – volume: 111 start-page: 673 year: 2017 ident: 743_CR11 publication-title: Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM doi: 10.1007/s13398-016-0318-y contributor: fullname: JA Conejero – volume-title: Lineability: The Search for Linearity in Mathematics, Monographs and Research Notes in Mathematics year: 2016 ident: 743_CR2 contributor: fullname: RM Aron – volume: 64 start-page: 539 year: 2015 ident: 743_CR9 publication-title: Mich. Math. J. doi: 10.1307/mmj/1441116657 contributor: fullname: G Botelho – volume: 48 start-page: 149 year: 2009 ident: 743_CR3 publication-title: Topology doi: 10.1016/j.top.2009.11.013 contributor: fullname: RM Aron – volume: 141 start-page: 1913 year: 2013 ident: 743_CR5 publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-2012-11466-2 contributor: fullname: CS Barroso – volume: 266 start-page: 3997 year: 2014 ident: 743_CR6 publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2013.11.014 contributor: fullname: L Bernal-González – volume: 237 start-page: 119 year: 2017 ident: 743_CR1 publication-title: Stud. 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| Title | Lineability and modes of convergence |
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