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 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Sierpiński-Zygmund functions and other problems on lineability

Author(s): José L. Gámez-Merino; Gustavo A. Muñoz-Fernández; Víctor M. Sánchez; Juan B. Seoane-Sepúlveda
Journal: Proc. Amer. Math. Soc. 138 (2010), 3863-3876.
MSC (2010): Primary 15A03, 26A15, 26A27, 46J10
Posted: May 24, 2010
MathSciNet review: 2679609
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We find large algebraic structures inside the following sets of pathological functions: (i) perfectly everywhere surjective functions, (ii) differentiable functions with almost nowhere continuous derivatives, (iii) differentiable nowhere monotone functions, and (iv) Sierpiński-Zygmund functions. The conclusions obtained on (i) and (iii) are improvements of some already known results.

References:

[1]
R. M. Aron, D. García, and M. Maestre, Linearity in non-linear problems, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 95 (2001), no. 1, 7-12. MR 1899348 (2003b:46062)

[2]
R. M. Aron, J. A. Conejero, A. Peris, and J. B. Seoane-Sepúlveda, Uncountably generated algebras of everywhere surjective functions, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 1-5.

[3]
R. M. Aron, F. J. García-Pacheco, D. Pérez-García, and J. B. Seoane-Sepúlveda, On dense-lineability of sets of functions on $ \mathbb{R}$, Topology 48 (2009), 149-156.

[4]
R. M. Aron, V. I. Gurariy, and J. B. Seoane-Sepúlveda, Lineability and spaceability of sets of functions on $ \mathbb{R}$, Proc. Amer. Math. Soc. 133 (2005), no. 3, 795-803. MR 2113929 (2006i:26004)

[5]
R. M. Aron, D. Pérez-García, and J. B. Seoane-Sepúlveda, Algebrability of the set of non-convergent Fourier series, Studia Math. 175 (2006), no. 1, 83-90. MR 2261701 (2007k:42007)

[6]
R. M. Aron and J. B. Seoane-Sepúlveda, Algebrability of the set of everywhere surjective functions on $ \mathbb{C}$, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 1, 25-31. MR 2327324 (2008d:26016)

[7]
F. Bayart and L. Quarta, Algebras in sets of queer functions, Israel J. Math. 158 (2007), 285-296. MR 2342549 (2008g:26006)

[8]
L. Bernal-González, Dense-lineability in spaces of continuous functions, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3163-3169. MR 2407080 (2009c:46038)

[9]
H. Blumberg, New properties of all real functions, Trans. Amer. Math. Soc. 82 (1922), 53-61.

[10]
Geraldo Botelho, Diogo Diniz, and Daniel Pellegrino, Lineability of the set of bounded linear non-absolutely summing operators, J. Math. Anal. Appl. 357 (2009), no. 1, 171-175. MR 2526816

[11]
G. Botelho, M. Matos, and D. Pellegrino, Lineability of summing sets of homogeneous polynomials, Linear Multilinear Algebra 58 (2010), 61-74.

[12]
Andrew M. Bruckner, Differentiation of real functions, Lecture Notes in Mathematics, vol. 659, Springer-Verlag, Berlin, 1978. MR 507448 (80h:26002)

[13]
James Foran, Fundamentals of real analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 144, Marcel Dekker Inc., New York, 1991. MR 1201817 (94e:00002)

[14]
D. García, B.C. Grecu, M. Maestre, and J. B. Seoane-Sepúlveda, Infinite dimensional Banach spaces of functions with nonlinear properties, Math. Nachr. 283 (2010), no. 5, 712-720.

[15]
Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in analysis, Dover Publications Inc., Mineola, NY, 2003. Corrected reprint of the second (1965) edition. MR 1996162

[16]
V. I. Gurariy, Subspaces and bases in spaces of continuous functions (Russian), Dokl. Akad. Nauk SSSR 167 (1966), 971-973. MR 0199674 (33:7817)

[17]
V. I. Gurariy, Linear spaces composed of nondifferentiable functions, C. R. Acad. Bulgare Sci. 44 (1991), 13-16.

[18]
A. B. Kharazishvili, Strange functions in real analysis, 2nd ed., Pure and Applied Mathematics, vol. 272, Chapman & Hall/CRC, Boca Raton, Florida, 2006. MR 2193523 (2006g:26003)

[19]
H. Lebesgue, Leçons sur l'intégration et la recherche des fonctions primitives, Gauthier-Villars, Paris, 1904.

[20]
Jaroslav Lukeš, Jan Malý, and Luděk Zajíček, Fine topology methods in real analysis and potential theory, Lecture Notes in Mathematics, vol. 1189, Springer-Verlag, Berlin, 1986. MR 861411 (89b:31001)

[21]
G. A. Muñoz-Fernández, N. Palmberg, D. Puglisi, and J. B. Seoane-Sepúlveda, Lineability in subsets of measure and function spaces, Linear Algebra Appl. 428 (2008), no. 11-12, 2805-2812. MR 2416590 (2009g:46051)

[22]
D. Puglisi and J. B. Seoane-Sepúlveda, Bounded linear non-absolutely summing operators, J. Math. Anal. Appl. 338 (2008), no. 1, 292-298. MR 2386416 (2009e:47038)

[23]
Juichi Shinoda, Some consequences of Martin's axiom and the negation of the continuum hypothesis, Nagoya Math. J. 49 (1973), 117-125. MR 0319754 (47:8296)

[24]
W. Sierpiński and A. Zygmund, Sur une fonction qui est discontinue sur tout ensemble de puissance du continu, Fund. Math. 4 (1923), 316-318.

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Additional Information:

José L. Gámez-Merino
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid 28040, Spain
Email: jlgamez@mat.ucm.es

Gustavo A. Muñoz-Fernández
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid 28040, Spain
Email: gustavo_fernandez@mat.ucm.es

Víctor M. Sánchez
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid 28040, Spain
Email: victorms@mat.ucm.es

Juan B. Seoane-Sepúlveda
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid 28040, Spain
Email: jseoane@mat.ucm.es

DOI: 10.1090/S0002-9939-2010-10420-3
PII: S 0002-9939(2010)10420-3
Keywords: Lineability, algebrability, everywhere surjective, nowhere monotone, approximately continuous, Pompeiu derivative.
Received by editor(s): September 17, 2009
Received by editor(s) in revised form: February 1, 2010
Posted: May 24, 2010
Additional Notes: The first, second and fourth authors were supported by the Spanish Ministry of Science and Innovation, grant MTM2009-07848.
The third author was supported by the Spanish Ministry of Science and Innovation, grant MTM2008-02652.
Dedicated: Dedicated to Professor Richard M. Aron on his 65$^{\text {th}}$ anniversary
Communicated by: Michael T. Lacey
Copyright of article: Copyright 2010, American Mathematical Society




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