Algebrability, non-linear properties, and special functions

Bartoszewicz, Artur; Gła̧b, Szymon; Pellegrino, Daniel; Seoane-Sepúlveda, Juan

doi:10.1090/S0002-9939-2013-11641-2

Algebrability, non-linear properties, and special functions
HTML articles powered by AMS MathViewer

by Artur Bartoszewicz, Szymon Gła̧b, Daniel Pellegrino and Juan B. Seoane-Sepúlveda PDF

Proc. Amer. Math. Soc. 141 (2013), 3391-3402 Request permission

Abstract:

We construct uncountably generated algebras inside the following sets of special functions: (i) Sierpiński-Zygmund functions, (ii) perfectly everywhere surjective functions, (iii) nowhere continuous Darboux functions. All conclusions obtained in this paper are improvements of some already known results.

References

Richard M. Aron, José A. Conejero, Alfredo Peris, and Juan B. Seoane-Sepúlveda, Uncountably generated algebras of everywhere surjective functions, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 3, 571–575. MR 2731374
Richard Aron, Domingo García, and Manuel Maestre, Linearity in non-linear problems, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 95 (2001), no. 1, 7–12 (English, with English and Spanish summaries). MR 1899348
R. M. Aron, F. J. García-Pacheco, D. Pérez-García, and J. B. Seoane-Sepúlveda, On dense-lineability of sets of functions on $\Bbb R$, Topology 48 (2009), no. 2-4, 149–156. MR 2596209, DOI 10.1016/j.top.2009.11.013
Richard Aron, V. I. Gurariy, and J. B. Seoane, Lineability and spaceability of sets of functions on $\Bbb R$, Proc. Amer. Math. Soc. 133 (2005), no. 3, 795–803. MR 2113929, DOI 10.1090/S0002-9939-04-07533-1
Richard M. Aron, David Pérez-García, and Juan B. Seoane-Sepúlveda, Algebrability of the set of non-convergent Fourier series, Studia Math. 175 (2006), no. 1, 83–90. MR 2261701, DOI 10.4064/sm175-1-5
Richard M. Aron and Juan B. Seoane-Sepúlveda, Algebrability of the set of everywhere surjective functions on $\Bbb C$, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 1, 25–31. MR 2327324
B. Balcar and F. Franěk, Independent families in complete Boolean algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 607–618. MR 675069, DOI 10.1090/S0002-9947-1982-0675069-3
Marek Balcerzak, Krzysztof Ciesielski, and Tomasz Natkaniec, Sierpiński-Zygmund functions that are Darboux, almost continuous, or have a perfect road, Arch. Math. Logic 37 (1997), no. 1, 29–35. MR 1485861, DOI 10.1007/s001530050080
A. Bartoszewicz and S. Gła̧b, Strong algebrability of sets of sequences and functions, Proc. Amer. Math. Soc. 141 (2013), 827–835., DOI 10.1090/S0002-9939-2012-11377-2
Artur Bartoszewicz and Szymon Gła̧b, Algebrability of conditionally convergent series with Cauchy product, J. Math. Anal. Appl. 385 (2012), no. 2, 693–697. MR 2834845, DOI 10.1016/j.jmaa.2011.07.008
Artur Bartoszewicz, Szymon Gła̧b, and Tadeusz Poreda, On algebrability of nonabsolutely convergent series, Linear Algebra Appl. 435 (2011), no. 5, 1025–1028. MR 2807216, DOI 10.1016/j.laa.2011.02.008
Frédéric Bayart and Lucas Quarta, Algebras in sets of queer functions, Israel J. Math. 158 (2007), 285–296. MR 2342549, DOI 10.1007/s11856-007-0014-x
L. Bernal-González, Dense-lineability in spaces of continuous functions, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3163–3169. MR 2407080, DOI 10.1090/S0002-9939-08-09495-1
Henry Blumberg, New properties of all real functions, Trans. Amer. Math. Soc. 24 (1922), no. 2, 113–128. MR 1501216, DOI 10.1090/S0002-9947-1922-1501216-9
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, DOI 10.1016/j.jmaa.2009.03.062
G. Botelho, V. V. Fávaro, D. Pellegrino, and J. B. Seoane-Sepúlveda, $L_p[0,1]\sbs \bigcup _{q>p}L_q[0,1]$ is spaceable for every $p>0$, Linear Algebra Appl. 436 (2012), no. 9, 2963–2965. MR 2900689, DOI 10.1016/j.laa.2011.12.028
Geraldo Botelho, Mário C. Matos, and Daniel Pellegrino, Lineability of summing sets of homogeneous polynomials, Linear Multilinear Algebra 58 (2010), no. 1-2, 61–74. MR 2641522, DOI 10.1080/03081080802095446
Krzysztof Ciesielski and Tomasz Natkaniec, Algebraic properties of the class of Sierpiński-Zygmund functions, Topology Appl. 79 (1997), no. 1, 75–99. MR 1462608, DOI 10.1016/S0166-8641(96)00128-9
Krzysztof Ciesielski and Tomasz Natkaniec, On Sierpiński-Zygmund bijections and their inverses, Topology Proc. 22 (1997), no. Spring, 155–164. MR 1657922
Per H. Enflo, Vladimir I. Gurariy, and J. B. Seoane-Sepúlveda, Some results and open questions on spaceability in function spaces, Trans. Amer. Math. Soc., in press.
P. Erdős, A. Hajnal, and E. C. Milner, On sets of almost disjoint subsets of a set, Acta Math. Acad. Sci. Hungar. 19 (1968), 209–218. MR 224485, DOI 10.1007/BF01894696
James Foran, Fundamentals of real analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 144, Marcel Dekker, Inc., New York, 1991. MR 1201817
José L. Gámez-Merino, Large algebraic structures inside the set of surjective functions, Bull. Belg. Math. Soc. Simon Stevin 18 (2011), no. 2, 297–300. MR 2848805
José L. Gámez-Merino, Gustavo A. Muñoz-Fernández, Daniel Pellegrino, and Juan B. Seoane-Sepúlveda, Bounded and unbounded polynomials and multilinear forms: characterizing continuity, Linear Algebra Appl. 436 (2012), no. 1, 237–242. MR 2859924, DOI 10.1016/j.laa.2011.06.050
J. L. Gámez-Merino, G. A. Muñoz-Fernández, D. Pellegrino, and J. B. Seoane-Sepúlveda, Lineability and algebrability in subsets of complex functions, preprint.
José L. Gámez-Merino, Gustavo A. Muñoz-Fernández, Víctor M. Sánchez, and Juan B. Seoane-Sepúlveda, Sierpiński-Zygmund functions and other problems on lineability, Proc. Amer. Math. Soc. 138 (2010), no. 11, 3863–3876. MR 2679609, DOI 10.1090/S0002-9939-2010-10420-3
José L. Gámez, Gustavo A. Muñoz-Fernández, and Juan B. Seoane-Sepúlveda, Lineability and additivity in $\Bbb R^{\Bbb R}$, J. Math. Anal. Appl. 369 (2010), no. 1, 265–272. MR 2643865, DOI 10.1016/j.jmaa.2010.03.036
José L. Gámez-Merino, Gustavo A. Muñoz-Fernández, and Juan B. Seoane-Sepúlveda, A characterization of continuity revisited, Amer. Math. Monthly 118 (2011), no. 2, 167–170. MR 2795585, DOI 10.4169/amer.math.monthly.118.02.167
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. MR 2666300, DOI 10.1002/mana.200610833
F. J. García-Pacheco, M. Martín, and J. B. Seoane-Sepúlveda, Lineability, spaceability, and algebrability of certain subsets of function spaces, Taiwanese J. Math. 13 (2009), no. 4, 1257–1269. MR 2543741, DOI 10.11650/twjm/1500405506
Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in analysis, The Mathesis Series, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964. MR 0169961
V. I. Gurariy, Subspaces and bases in spaces of continuous functions (Russian), Dokl. Akad. Nauk SSSR 167 (1966), 971–973.
V. I. Gurariy, Linear spaces composed of nondifferentiable functions, C.R. Acad. Bulgare Sci. 44 (1991), 13–16.
Stanislav Hencl, Isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3505–3511. MR 1707147, DOI 10.1090/S0002-9939-00-05595-7
F. B. Jones, Connected and disconnected plane sets and the functional equation $f(x)+f(y)=f(x+y)$, Bull. Amer. Math. Soc. 48 (1942), 115–120. MR 5906, DOI 10.1090/S0002-9904-1942-07615-4
A. B. Kharazishvili, Strange functions in real analysis, 2nd ed., Pure and Applied Mathematics (Boca Raton), vol. 272, Chapman & Hall/CRC, Boca Raton, FL, 2006. MR 2193523
Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
Henri Leon Lebesgue, Leçons sur l’intégration et la recherche des fonctions primitives professées au Collège de France, Cambridge Library Collection, Cambridge University Press, Cambridge, 2009 (French). Reprint of the 1904 original. MR 2857993, DOI 10.1017/CBO9780511701825
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, DOI 10.1016/j.laa.2008.01.008
Krzysztof Płotka, Sum of Sierpiński-Zygmund and Darboux like functions, Topology Appl. 122 (2002), no. 3, 547–564. MR 1911699, DOI 10.1016/S0166-8641(01)00184-5
Juichi Shinoda, Some consequences of Martin’s axiom and the negation of the continuum hypothesis, Nagoya Math. J. 49 (1973), 117–125. MR 319754, DOI 10.1017/S0027763000015336
Juan B. Seoane, Chaos and lineability of pathological phenomena in analysis, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–Kent State University. MR 2709064
W. Sierpiński and A. Zygmund, Sur une fonction qui est discontinue sur tout ensemble de puissance du continu, Fund. Math. 4 (1923), 316–318., DOI 10.4064/fm-4-1-316-318