Lineability and K-linear discontinuous functions
It is well known that there are Q-linear discontinuous functions from R to R. Moreover, such functions have dense graph in the real plane. In this paper, we study the existence of linear spaces contained in the family of linear discontinuous functions in the generalized context of topological vector...
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| Published in: | Linear algebra and its applications Vol. 645; pp. 52 - 67 |
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| Abstract | It is well known that there are Q-linear discontinuous functions from R to R. Moreover, such functions have dense graph in the real plane. In this paper, we study the existence of linear spaces contained in the family of linear discontinuous functions in the generalized context of topological vector spaces and fields endowed with an absolute value. Furthermore, in the case of fields endowed with an absolute value, we study the algebraic genericity of the families of (1) linear discontinuous functions that are not everywhere surjective and (2) linear discontinuous functions that are strongly everywhere surjective. This extends a number of previous results of lineability theory.
Generalizations of classical results of Q-linear discontinuous functions are also analyzed. |
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| AbstractList | It is well known that there are Q-linear discontinuous functions from R to R. Moreover, such functions have dense graph in the real plane. In this paper, we study the existence of linear spaces contained in the family of linear discontinuous functions in the generalized context of topological vector spaces and fields endowed with an absolute value. Furthermore, in the case of fields endowed with an absolute value, we study the algebraic genericity of the families of (1) linear discontinuous functions that are not everywhere surjective and (2) linear discontinuous functions that are strongly everywhere surjective. This extends a number of previous results of lineability theory.
Generalizations of classical results of Q-linear discontinuous functions are also analyzed. It is well known that there are R to R. Moreover, such functions have dense graph in the real plane. In this paper, we study the existence of linear spaces contained in the family of linear discontinuous functions in the generalized context of topological vector spaces and fields endowed with an absolute value. Furthermore, in the case of fields endowed with an absolute value, we study the algebraic genericity of the families of (1) linear discontinuous functions that are not everywhere surjective and (2) linear discontinuous functions that are strongly everywhere surjective. This extends a number of previous results of lineability theory. |
| Author | Rodríguez-Vidanes, D.L. Seoane-Sepúlveda, J.B. Trutschnig, W. Fernández-Sánchez, J. |
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| Cites_doi | 10.4064/sm-5-1-69-98 10.1016/j.jmaa.2006.03.025 10.1090/S0002-9939-2010-10420-3 10.1016/j.laa.2006.02.041 10.7146/math.scand.a-15057 10.1016/j.jmaa.2004.01.036 10.1090/S0273-0979-2013-01421-6 10.1515/math-2017-0042 10.1090/S0002-9939-04-07533-1 10.1090/bull/1635 10.1090/S0002-9904-1942-07615-4 |
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| Keywords | Lineability 12J99 Linear function Strongly everywhere surjective 46S10 46B87 Normed space 15A03 Field Everywhere surjective Topological vector space Discontinuous function Absolute value |
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| References | Fernández-Sánchez, Maghsoudi, Rodríguez-Vidanes, Seoane-Sepúlveda (br0100) 2022; 116 Gámez-Merino, Muñoz Fernández, Sánchez, Seoane-Sepúlveda (br0190) 2010; 138 Ciesielski, Seoane-Sepúlveda (br0080) 2019; 56 Gurariy, Quarta (br0090) 2004; 294 Fenoy-Muñoz, Gámez-Merino, Muñoz Fernández, Sáez-Maestro (br0120) 2017; 15 Aczél, Dhombres (br0200) 1989; vol. 31 Robert (br0130) 2000; vol. 198 Jech (br0170) 2003 Köthe (br0150) 1979; vol. 237 Bernal-González, Pellegrino, Seoane-Sepúlveda (br0050) 2014; 51 García-Pacheco, Palmberg, Seoane-Sepúlveda (br0010) 2007; 326 Aron, Gurariy, Seoane (br0040) 2005; 133 Köthe (br0140) 1969; Band 159 Jones (br0020) 1942; 48 Seoane-Sepúlveda (br0070) 2006 Aizpuru, Pérez-Eslava, Seoane-Sepúlveda (br0030) 2006; 418 Hodel (br0160) 1984 García-Pacheco, Rambla-Barreno, Seoane-Sepúlveda (br0110) 2008; 102 Aron, Bernal González, Pellegrino, Seoane Sepúlveda (br0060) 2016 Fichtenholz, Kantorovich (br0180) 1934; 5 Gámez-Merino (10.1016/j.laa.2022.03.007_br0190) 2010; 138 Robert (10.1016/j.laa.2022.03.007_br0130) 2000; vol. 198 Aczél (10.1016/j.laa.2022.03.007_br0200) 1989; vol. 31 Aizpuru (10.1016/j.laa.2022.03.007_br0030) 2006; 418 Ciesielski (10.1016/j.laa.2022.03.007_br0080) 2019; 56 Jones (10.1016/j.laa.2022.03.007_br0020) 1942; 48 Gurariy (10.1016/j.laa.2022.03.007_br0090) 2004; 294 Jech (10.1016/j.laa.2022.03.007_br0170) 2003 Fichtenholz (10.1016/j.laa.2022.03.007_br0180) 1934; 5 García-Pacheco (10.1016/j.laa.2022.03.007_br0010) 2007; 326 García-Pacheco (10.1016/j.laa.2022.03.007_br0110) 2008; 102 Fenoy-Muñoz (10.1016/j.laa.2022.03.007_br0120) 2017; 15 Köthe (10.1016/j.laa.2022.03.007_br0150) 1979; vol. 237 Bernal-González (10.1016/j.laa.2022.03.007_br0050) 2014; 51 Fernández-Sánchez (10.1016/j.laa.2022.03.007_br0100) 2022; 116 Aron (10.1016/j.laa.2022.03.007_br0040) 2005; 133 Köthe (10.1016/j.laa.2022.03.007_br0140) 1969; Band 159 Aron (10.1016/j.laa.2022.03.007_br0060) 2016 Seoane-Sepúlveda (10.1016/j.laa.2022.03.007_br0070) 2006 Hodel (10.1016/j.laa.2022.03.007_br0160) 1984 |
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| SubjectTerms | Absolute value Discontinuous function Everywhere surjective Field Lineability Linear algebra Linear function Normed space Strongly everywhere surjective Topological vector space Vector spaces |
| Title | Lineability and K-linear discontinuous functions |
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