Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Linear subsets of nonlinear sets in topological vector spaces


Authors: Luis Bernal-González, Daniel Pellegrino and Juan B. Seoane-Sepúlveda
Journal: Bull. Amer. Math. Soc. 51 (2014), 71-130
MSC (2010): Primary 15A03, 46E10, 46E15; Secondary 26B05, 28A20, 47A16, 47L05
DOI: https://doi.org/10.1090/S0273-0979-2013-01421-6
Published electronically: July 15, 2013
MathSciNet review: 3119823
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability.


References [Enhancements On Off] (What's this?)

  • [1] María D. Acosta, Antonio Aizpuru, Richard M. Aron, and Francisco J. García-Pacheco, Functionals that do not attain their norm, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 3, 407-418. MR 2387038 (2009i:46021)
  • [2] A. Aizpuru, C. Pérez-Eslava, F. J. García-Pacheco, and J. B. Seoane-Sepúlveda, Lineability and coneability of discontinuous functions on $ \mathbb{R}$, Publ. Math. Debrecen 72 (2008), no. 1-2, 129-139. MR 2376864 (2008m:26002)
  • [3] A. Aizpuru, C. Pérez-Eslava, and J. B. Seoane-Sepúlveda, Linear structure of sets of divergent sequences and series, Linear Algebra Appl. 418 (2006), no. 2-3, 595-598. MR 2260214 (2008h:40001), https://doi.org/10.1016/j.laa.2006.02.041
  • [4] I. Akbarbaglu and S. Maghsoudi, Large structures in certain subsets of Orlicz spaces, Linear Algebra Appl. 438 (2013), no. 11, 4363-4373. MR 3034537, https://doi.org/10.1016/j.laa.2013.01.038
  • [5] I. Akbarbaglu, S. Maghsoudi, and J. B. Seoane-Sepúlveda, Porous sets and lineability of continuous functions on locally compact groups, J. Math. Anal. Appl. 406 (2013), no. 1, 211-218.
  • [6] A. Albanese, X. Barrachina, E. M. Mangino, and A. Peris, Distributional chaos for strongly continuous semigroups of operators, Comm. Pure Appl. Analysis 12 (2013), no. 5, 2069-2082.
  • [7] Pieter C. Allaart and Kiko Kawamura, The improper infinite derivatives of Takagi's nowhere-differentiable function, J. Math. Anal. Appl. 372 (2010), no. 2, 656-665. MR 2678891 (2011k:26004), https://doi.org/10.1016/j.jmaa.2010.06.059
  • [8] Shamim I. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), no. 2, 384-390. MR 1469346 (98h:47028a), https://doi.org/10.1006/jfan.1996.3093
  • [9] D. H. Armitage, A non-constant continuous function on the plane whose integral on every line is zero, Amer. Math. Monthly 101 (1994), no. 9, 892-894. MR 1300495 (95m:30053), https://doi.org/10.2307/2975138
  • [10] David H. Armitage, Dense vector spaces of universal harmonic functions, Advances in multivariate approximation (Witten-Bommerholz, 1998) Math. Res., vol. 107, Wiley-VCH, Berlin, 1999, pp. 33-42. MR 1797219 (2001k:31004)
  • [11] D. H. Armitage, Entire functions decaying rapidly on strips, Quaest. Math. 23 (2000), no. 4, 417-424. MR 1810291 (2001i:30024), https://doi.org/10.2989/16073600009485988
  • [12] Richard Aron, An introduction to polynomials on Banach spaces, Extracta Math. 17 (2002), no. 3, 303-329. IV Course on Banach Spaces and Operators (Spanish) (Laredo, 2001). MR 1995411 (2004f:46053)
  • [13] Richard M. Aron, Linearity in non-linear situations, Advanced courses of mathematical analysis. II, World Sci. Publ., Hackensack, NJ, 2007, pp. 1-15. MR 2334322 (2008f:46001), https://doi.org/10.1142/9789812708441_0001
  • [14] R. Aron, J. Bès, F. León, and A. Peris, Operators with common hypercyclic subspaces, J. Operator Theory 54 (2005), no. 2, 251-260. MR 2186352 (2006h:47010)
  • [15] R. M. Aron, C. Boyd, R. A. Ryan, and I. Zalduendo, Zeros of polynomials on Banach spaces: the real story, Positivity 7 (2003), no. 4, 285-295. MR 2017308 (2004j:46066), https://doi.org/10.1023/A:1026278115574
  • [16] R. M. Aron, J. A. Conejero, A. Peris, and J. B. Seoane-Sepúlveda, Sums and products of bad functions, Function spaces, Contemp. Math., vol. 435, Amer. Math. Soc., Providence, RI, 2007, pp. 47-52. MR 2359417 (2008m:46062), https://doi.org/10.1090/conm/435/08365
  • [17] R. M. Aron, J. A. Conejero, A. Peris, and J. B. Seoane-Sepúlveda, Powers of hypercyclic functions for some classical hypercyclic operators, Integral Equations Operator Theory 58 (2007), no. 4, 591-596. MR 2329137 (2008e:47017), https://doi.org/10.1007/s00020-007-1490-4
  • [18] 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 (2011g:46041)
  • [19] 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 (2003b:46062)
  • [20] 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), no. 2-4, 149-156. MR 2596209 (2011c:26011), https://doi.org/10.1016/j.top.2009.11.013
  • [21] Richard Aron, Raquel Gonzalo, and Andriy Zagorodnyuk, Zeros of real polynomials, Linear and Multilinear Algebra 48 (2000), no. 2, 107-115. MR 1813438 (2001k:12002), https://doi.org/10.1080/03081080008818662
  • [22] Richard 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 (electronic). MR 2113929 (2006i:26004), https://doi.org/10.1090/S0002-9939-04-07533-1
  • [23] Richard M. Aron and Petr Hájek, Odd degree polynomials on real Banach spaces, Positivity 11 (2007), no. 1, 143-153. MR 2297328 (2007m:46064), https://doi.org/10.1007/s11117-006-2035-9
  • [24] R. M. Aron and P. Hájek, Zero sets of polynomials in several variables, Arch. Math. (Basel) 86 (2006), no. 6, 561-568. MR 2241604 (2008c:46063), https://doi.org/10.1007/s00013-006-1314-9
  • [25] 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 (2007k:42007), https://doi.org/10.4064/sm175-1-5
  • [26] R. M. Aron and M. P. Rueda, A problem concerning zero-subspaces of homogeneous polynomials, Linear Topol. Spaces Complex Anal. 3 (1997), 20-23. Dedicated to Professor Vyacheslav Pavlovich Zahariuta. MR 1632480 (99e:13032)
  • [27] Richard M. Aron and Juan 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)
  • [28] Antonio Avilés and Stevo Todorcevic, Zero subspaces of polynomials on $ l_1(\Gamma )$, J. Math. Anal. Appl. 350 (2009), no. 2, 427-435. MR 2474778 (2009k:46077), https://doi.org/10.1016/j.jmaa.2007.08.020
  • [29] D. Azagra, G. A. Muñoz-Fernández, V. M. Sánchez, and J. B. Seoane-Sepúlveda, Riemann integrability and Lebesgue measurability of the composite function, J. Math. Anal. Appl. 354 (2009), no. 1, 229-233. MR 2510434 (2010i:26016), https://doi.org/10.1016/j.jmaa.2008.12.033
  • [30] A.G. Bacharoglu, Universal Taylor series on doubly connected domains, Results in Maths. 53 (2009), no. 1-2, 9-18.
  • [31] M. Balcerzak, A. Bartoszewicz, and M. Filipczac, Nonseparable spaceability and strong algebrability of sets of continuous singular functions, J. Math. Anal. Appl. (to appear).
  • [32] 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 (98k:26005), https://doi.org/10.1007/s001530050080
  • [33] T. Banakh, A. Bartoszewicz, S. Głab, and E. Szymonik, Algebrability, lineability and the subsums of series, Colloq. Math. 129 (2012), no. 3, 75-85.
  • [34] T. Banakh, A. Plichko, and A. Zagorodnyuk, Zeros of quadratic functionals on non-separable spaces, Colloq. Math. 100 (2004), no. 1, 141-147. MR 2079354 (2005g:46047), https://doi.org/10.4064/cm100-1-13
  • [35] Pradipta Bandyopadhyay and Gilles Godefroy, Linear structures in the set of norm-attaining functionals on a Banach space, J. Convex Anal. 13 (2006), no. 3-4, 489-497. MR 2291549 (2007k:46020)
  • [36] Cleon S. Barroso, Geraldo Botelho, Vinícius V. Fávaro, and Daniel Pellegrino, Lineability and spaceability for the weak form of Peano's theorem and vector-valued sequence spaces, Proc. Amer. Math. Soc. 141 (2013), no. 6, 1913-1923. MR 3034418, https://doi.org/10.1090/S0002-9939-2012-11466-2
  • [37] Artur Bartoszewicz, Marek Bienias, and Szymon Głab, Independent Bernstein sets and algebraic constructions, J. Math. Anal. Appl. 393 (2012), no. 1, 138-143. MR 2921655, https://doi.org/10.1016/j.jmaa.2012.03.007
  • [38] Artur Bartoszewicz and Szymon Głab, Algebrability of conditionally convergent series with Cauchy product, J. Math. Anal. Appl. 385 (2012), no. 2, 693-697. MR 2834845 (2012g:40001), https://doi.org/10.1016/j.jmaa.2011.07.008
  • [39] Artur Bartoszewicz and Szymon Głab, Strong algebrability of sets of sequences and functions, Proc. Amer. Math. Soc. 141 (2013), no. 3, 827-835. MR 3003676, https://doi.org/10.1090/S0002-9939-2012-11377-2
  • [40] A. Bartoszewicz, S. Głab, D. Pellegrino, and J.B. Seoane-Sepúlveda, Algebrability, nonlinear properties, and special functions, Proc. Amer. Math. Soc., in press.
  • [41] Artur Bartoszewicz, Szymon Głab, and Tadeusz Poreda, On algebrability of nonabsolutely convergent series, Linear Algebra Appl. 435 (2011), no. 5, 1025-1028. MR 2807216 (2012e:40001), https://doi.org/10.1016/j.laa.2011.02.008
  • [42] Françoise Bastin, Céline Esser, and Samuel Nicolay, Prevalence of ``nowhere analyticity'', Studia Math. 210 (2012), no. 3, 239-246. MR 2983462, https://doi.org/10.4064/sm210-3-4
  • [43] Frédéric Bayart, Linearity of sets of strange functions, Michigan Math. J. 53 (2005), no. 2, 291-303. MR 2152701 (2006b:46048), https://doi.org/10.1307/mmj/1123090769
  • [44] Frédéric Bayart, Topological and algebraic genericity of divergence and universality, Studia Math. 167 (2005), no. 2, 161-181. MR 2134382 (2006b:46024), https://doi.org/10.4064/sm167-2-4
  • [45] Frédéric Bayart, Common hypercyclic subspaces, Integral Equations Operator Theory 53 (2005), no. 4, 467-476. MR 2187432 (2006k:47016), https://doi.org/10.1007/s00020-004-1316-6
  • [46] Frédéric Bayart, Porosity and hypercyclic operators, Proc. Amer. Math. Soc. 133 (2005), no. 11, 3309-3316 (electronic). MR 2161154 (2006f:47008), https://doi.org/10.1090/S0002-9939-05-07842-1
  • [47] Frédéric Bayart and Sophie Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358 (2006), no. 11, 5083-5117 (electronic). MR 2231886 (2007e:47013), https://doi.org/10.1090/S0002-9947-06-04019-0
  • [48] Frédéric Bayart and Sophie Grivaux, Invariant Gaussian measures for operators on Banach spaces and linear dynamics, Proc. Lond. Math. Soc. (3) 94 (2007), no. 1, 181-210. MR 2294994 (2008i:47019), https://doi.org/10.1112/plms/pdl013
  • [49] F. Bayart, K.-G. Grosse-Erdmann, V. Nestoridis, and C. Papadimitropoulos, Abstract theory of universal series and applications, Proc. Lond. Math. Soc. (3) 96 (2008), no. 2, 417-463. MR 2396846 (2009j:30006), https://doi.org/10.1112/plms/pdm043
  • [50] F. Bayart, S. V. Konyagin, and H. Queffélec, Convergence almost everywhere and divergence everywhere of Taylor and Dirichlet series, Real Anal. Exchange 29 (2003/04), no. 2, 557-586. MR 2083797 (2005e:42026)
  • [51] F. Bayart and É. Matheron, Hypercyclic operators failing the hypercyclicity criterion on classical Banach spaces, J. Funct. Anal. 250 (2007), no. 2, 426-441. MR 2352487 (2008k:47016), https://doi.org/10.1016/j.jfa.2007.05.001
  • [52] Frédéric Bayart and Étienne Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, vol. 179, Cambridge University Press, Cambridge, 2009. MR 2533318 (2010m:47001)
  • [53] Frédéric Bayart and Vassili Nestoridis, Universal Taylor series have a strong form of universality, J. Anal. Math. 104 (2008), 69-82. MR 2403430 (2009e:30003), https://doi.org/10.1007/s11854-008-0017-5
  • [54] Frédéric Bayart and Lucas Quarta, Algebras in sets of queer functions, Israel J. Math. 158 (2007), 285-296. MR 2342549 (2008g:26006), https://doi.org/10.1007/s11856-007-0014-x
  • [55] Bernard Beauzamy, Un opérateur, sur l'espace de Hilbert, dont tous les polynômes sont hypercycliques, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 18, 923-925 (French, with English summary). MR 873395 (88g:47010)
  • [56] Bernard Beauzamy, Introduction to operator theory and invariant subspaces, North-Holland Mathematical Library, vol. 42, North-Holland Publishing Co., Amsterdam, 1988. MR 967989 (90d:47001)
  • [57] T. Bermúdez, A. Bonilla, F. Martínez-Giménez, and A. Peris, Li-Yorke and distributionally chaotic operators, J. Math. Anal. Appl. 373 (2011), no. 1, 83-93. MR 2684459 (2011j:47031), https://doi.org/10.1016/j.jmaa.2010.06.011
  • [58] Luis Bernal-González, A lot of ``counterexamples'' to Liouville's theorem, J. Math. Anal. Appl. 201 (1996), no. 3, 1002-1009. MR 1400577 (97d:30036), https://doi.org/10.1006/jmaa.1996.0298
  • [59] Luis Bernal-González, Small entire functions with extremely fast growth, J. Math. Anal. Appl. 207 (1997), no. 2, 541-548. MR 1438930 (98b:30028), https://doi.org/10.1006/jmaa.1997.5312
  • [60] Luis Bernal-González, On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1003-1010. MR 1476119 (99f:47010), https://doi.org/10.1090/S0002-9939-99-04657-2
  • [61] Luis Bernal-González, Densely hereditarily hypercyclic sequences and large hypercyclic manifolds, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3279-3285. MR 1646318 (2000b:47047), https://doi.org/10.1090/S0002-9939-99-05185-0
  • [62] Luis Bernal-González, Hypercyclic sequences of differential and antidifferential operators, J. Approx. Theory 96 (1999), no. 2, 323-337. MR 1671201 (2000b:47081), https://doi.org/10.1006/jath.1998.3237
  • [63] Luis Bernal-González, Universal images of universal elements, Studia Math. 138 (2000), no. 3, 241-250. MR 1758857 (2001f:47002)
  • [64] L. Bernal-González, Linear Kierst-Szpilrajn theorems, Studia Math. 166 (2005), no. 1, 55-69. MR 2108318 (2006m:30100), https://doi.org/10.4064/sm166-1-4
  • [65] L. Bernal-González, Hypercyclic subspaces in Fréchet spaces, Proc. Amer. Math. Soc. 134 (2006), no. 7, 1955-1961. MR 2215764 (2007a:47008), https://doi.org/10.1090/S0002-9939-05-08242-0
  • [66] L. Bernal-González, Linear structure of weighted holomorphic non-extendibility, Bull. Austral. Math. Soc. 73 (2006), no. 3, 335-344. MR 2230644 (2007b:30068), https://doi.org/10.1017/S0004972700035371
  • [67] L. Bernal-González, Dense-lineability in spaces of continuous functions, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3163-3169. MR 2407080 (2009c:46038), https://doi.org/10.1090/S0002-9939-08-09495-1
  • [68] L. Bernal-González, Lineability of sets of nowhere analytic functions, J. Math. Anal. Appl. 340 (2008), no. 2, 1284-1295. MR 2390929 (2009d:46047), https://doi.org/10.1016/j.jmaa.2007.09.048
  • [69] Luis Bernal-González, Algebraic genericity of strict-order integrability, Studia Math. 199 (2010), no. 3, 279-293. MR 2678906 (2011g:46049), https://doi.org/10.4064/sm199-3-5
  • [70] Luis Bernal-González, Lineability of universal divergence of Fourier series, Integral Equations Operator Theory 74 (2012), no. 2, 271-279. MR 2983066, https://doi.org/10.1007/s00020-012-1984-6
  • [71] Luis Bernal-González and Antonio Bonilla, Families of strongly annular functions: linear structure, Rev. Mat. Complut. 26 (2013), no. 1, 283-297. MR 3016629, https://doi.org/10.1007/s13163-011-0080-9
  • [72] L. Bernal-González, A. Bonilla, M. C. Calderón-Moreno, and J. A. Prado-Bassas, Maximal cluster sets of $ L$-analytic functions along arbitrary curves, Constr. Approx. 25 (2007), no. 2, 211-219. MR 2283498 (2007k:30055), https://doi.org/10.1007/s00365-006-0636-5
  • [73] L. Bernal-González, A. Bonilla, M. C. Calderón-Moreno, and J. A. Prado-Bassas, Universal Taylor series with maximal cluster sets, Rev. Mat. Iberoam. 25 (2009), no. 2, 757-780. MR 2569553 (2011a:30004), https://doi.org/10.4171/RMI/582
  • [74] Luis Bernal-González and M. D. C. Calderón-Moreno, Dense linear manifolds of monsters, J. Approx. Theory 119 (2002), no. 2, 156-180. MR 1939280 (2003h:30042), https://doi.org/10.1006/jath.2002.3712
  • [75] L. Bernal-González, M. C. Calderón-Moreno, and W. Luh, Large linear manifolds of noncontinuable boundary-regular holomorphic functions, J. Math. Anal. Appl. 341 (2008), no. 1, 337-345. MR 2394088 (2009b:30068), https://doi.org/10.1016/j.jmaa.2007.10.014
  • [76] L. Bernal-González, M. C. Calderón-Moreno, and W. Luh, Dense-lineability of sets of Birkhoff-universal functions with rapid decay, J. Math. Anal. Appl. 363 (2010), no. 1, 327-335. MR 2559068 (2010k:30029), https://doi.org/10.1016/j.jmaa.2009.08.049
  • [77] L. Bernal-González, M. C. Calderón-Moreno, and J. A. Prado-Bassas, Maximal cluster sets along arbitrary curves, J. Approx. Theory 129 (2004), no. 2, 207-216. MR 2078649 (2005f:30067), https://doi.org/10.1016/j.jat.2004.06.003
  • [78] L. Bernal-González, M. C. Calderón-Moreno, and J. A. Prado-Bassas, Cyclicity of coefficient multipliers: linear structure, Acta Math. Hungar. 114 (2007), no. 4, 287-300. MR 2305820 (2008d:47017), https://doi.org/10.1007/s10474-007-5125-7
  • [79] L. Bernal-González, M. C. Calderón-Moreno, and J. A. Prado-Bassas, Holomorphic operators generating dense images, Integral Equations Operator Theory 60 (2008), no. 1, 1-11. MR 2380312 (2009e:47022), https://doi.org/10.1007/s00020-007-1547-4
  • [80] L. Bernal-González, M. C. Calderón-Moreno, and J. A. Prado-Bassas, Large subspaces of compositionally universal functions with maximal cluster sets, J. Approx. Theory 164 (2012), no. 2, 253-267. MR 2864645, https://doi.org/10.1016/j.jat.2011.10.005
  • [81] L. Bernal-González and K.-G. Grosse-Erdmann, The hypercyclicity criterion for sequences of operators, Studia Math. 157 (2003), no. 1, 17-32. MR 1980114 (2003m:47013), https://doi.org/10.4064/sm157-1-2
  • [82] Luis Bernal González and Alfonso Montes-Rodríguez, Universal functions for composition operators, Complex Variables Theory Appl. 27 (1995), no. 1, 47-56. MR 1316270 (96a:30041)
  • [83] Luis Bernal González and Alfonso Montes Rodríguez, Non-finite-dimensional closed vector spaces of universal functions for composition operators, J. Approx. Theory 82 (1995), no. 3, 375-391. MR 1348728 (96f:30034), https://doi.org/10.1006/jath.1995.1086
  • [84] Luis Bernal-González and Manuel Ordóñez Cabrera, Spaceability of strict order integrability, J. Math. Anal. Appl. 385 (2012), no. 1, 303-309. MR 2834257 (2012g:46025), https://doi.org/10.1016/j.jmaa.2011.06.043
  • [85] L. Bernal-González and J. A. Prado-Tendero, U-operators, J. Aust. Math. Soc. 78 (2005), no. 1, 59-89. MR 2129489 (2006c:30039), https://doi.org/10.1017/S1446788700015561
  • [86] N. Bernardes, A. Bonilla, V. Müller, and A. Peris, Li-Yorke chaos in Linear Dynamics, Preprint (2012).
  • [87] Juan P. Bès, Invariant manifolds of hypercyclic vectors for the real scalar case, Proc. Amer. Math. Soc. 127 (1999), no. 6, 1801-1804. MR 1485460 (99i:47002), https://doi.org/10.1090/S0002-9939-99-04720-6
  • [88] J.P. Bès, Dynamics of composition operators with holomorphic symbol, Rev. Real Acad. Cien. Ser. A Mat., in press.
  • [89] Juan Bès and José A. Conejero, Hypercyclic subspaces in omega, J. Math. Anal. Appl. 316 (2006), no. 1, 16-23. MR 2201746 (2007b:47016), https://doi.org/10.1016/j.jmaa.2005.04.083
  • [90] Juan Bès and Alfredo Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), no. 1, 94-112. MR 1710637 (2000f:47012), https://doi.org/10.1006/jfan.1999.3437
  • [91] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164. MR 0115069 (22 #5872)
  • [92] G.D. Birkhoff, Démonstration d'un théorème élémentaire sur les fonctions entières, C. R. Acad. Sci. Paris 189 (1929), 473-475.
  • [93] Errett Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67 (1961), 97-98. MR 0123174 (23 #A503)
  • [94] H. Blumberg, New properties of all real functions, Trans. Amer. Math. Soc. 82 (1922), 53-61.
  • [95] P. du Bois-Reymond, Über den Gültigkeitsbereich der Taylorschen Reihenentwicklung, Math. Ann. 21 (1876), 109-117.
  • [96] D. D. Bonar and F. W. Carroll, Annular functions form a residual set, J. Reine Angew. Math. 272 (1975), 23-24. MR 0417428 (54 #5478)
  • [97] J. Bonet, F. Martínez-Giménez, and A. Peris, Linear chaos on Fréchet spaces, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), no. 7, 1649-1655. Dynamical systems and functional equations (Murcia, 2000). MR 2015614 (2004i:47016), https://doi.org/10.1142/S0218127403007497
  • [98] José Bonet, Félix Martínez-Giménez, and Alfredo Peris, Universal and chaotic multipliers on spaces of operators, J. Math. Anal. Appl. 297 (2004), no. 2, 599-611. Special issue dedicated to John Horváth. MR 2088683 (2005g:47006), https://doi.org/10.1016/j.jmaa.2004.03.073
  • [99] José Bonet and Alfredo Peris, Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal. 159 (1998), no. 2, 587-595. MR 1658096 (99k:47044), https://doi.org/10.1006/jfan.1998.3315
  • [100] A. Bonilla, ``Counterexamples'' to the harmonic Liouville theorem and harmonic functions with zero nontangential limits, Colloq. Math. 83 (2000), no. 2, 155-160. MR 1758311 (2001d:31003)
  • [101] A. Bonilla, Small entire functions with infinite growth index, J. Math. Anal. Appl. 267 (2002), no. 1, 400-404. MR 1886838 (2003d:30031), https://doi.org/10.1006/jmaa.2001.7767
  • [102] A. Bonilla, Universal harmonic functions, Quaest. Math. 25 (2002), 527-530.
  • [103] A. Bonilla and K.-G. Grosse-Erdmann, On a theorem of Godefroy and Shapiro, Integral Equations Operator Theory 56 (2006), no. 2, 151-162. MR 2264513 (2007h:47020), https://doi.org/10.1007/s00020-006-1423-7
  • [104] A. Bonilla and K.-G. Grosse-Erdmann, Frequently hypercyclic operators and vectors, Ergodic Theory Dynam. Systems 27 (2007), no. 2, 383-404. MR 2308137 (2008c:47016), https://doi.org/10.1017/S014338570600085X
  • [105] A. Bonilla and K.-G. Grosse-Erdmann, Frequently hypercyclic subspaces, Monatsh. Math. 168 (2012), no. 3-4, 305-320. MR 2993952, https://doi.org/10.1007/s00605-011-0369-2
  • [106] Jonathan M. Borwein and Xianfu Wang, Lipschitz functions with maximal Clarke subdifferentials are staunch, Bull. Austral. Math. Soc. 72 (2005), no. 3, 491-496. MR 2199651 (2007a:49024), https://doi.org/10.1017/S0004972700035322
  • [107] Geraldo Botelho, Daniel Cariello, Vinícius V. Fávaro, and Daniel Pellegrino, Maximal spaceability in sequence spaces, Linear Algebra Appl. 437 (2012), no. 12, 2978-2985. MR 2966612, https://doi.org/10.1016/j.laa.2012.06.043
  • [108] G. Botelho, D. Cariello, V.V. Fávaro, D. Pellegrino, and J.B. Seoane-Sepúlveda, Distinguished subspaces of $ L_p$ of maximal dimension, Studia Math. (to appear).
  • [109] G. Botelho, D. Diniz, V. V. Fávaro, and D. Pellegrino, Spaceability in Banach and quasi-Banach sequence spaces, Linear Algebra Appl. 434 (2011), no. 5, 1255-1260. MR 2763584 (2011m:46006), https://doi.org/10.1016/j.laa.2010.11.012
  • [110] 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 (2010d:47025), https://doi.org/10.1016/j.jmaa.2009.03.062
  • [111] G. Botelho, V. V. Fávaro, D. Pellegrino, and J. B. Seoane-Sepúlveda, $ L_p[0,1]\backslash \bigcup _{q>p}L_q[0,1]$ is spaceable for every $ p>0$, Linear Algebra Appl. 436 (2012), no. 9, 2963-2965. MR 2900689, https://doi.org/10.1016/j.laa.2011.12.028
  • [112] 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 (2011b:46072), https://doi.org/10.1080/03081080802095446
  • [113] Geraldo Botelho, Daniel Pellegrino, and Pilar Rueda, Cotype and absolutely summing linear operators, Math. Z. 267 (2011), no. 1-2, 1-7. MR 2772238 (2012e:46022), https://doi.org/10.1007/s00209-009-0591-y
  • [114] Paul S. Bourdon, Invariant manifolds of hypercyclic vectors, Proc. Amer. Math. Soc. 118 (1993), no. 3, 845-847. MR 1148021 (93i:47002), https://doi.org/10.2307/2160131
  • [115] P.S. Bourdon and J.H. Shapiro, The role of the spectrum in the cyclic behavior of composition operators, Memoirs Amer. Math. Soc. 596, AMS, Providence, Rhode Island, 1997.
  • [116] J. Bourgain, On the distribution of Dirichlet sums, J. Anal. Math. 60 (1993), 21-32.
  • [117] Andrew M. Bruckner, Differentiation of real functions, Lecture Notes in Mathematics, vol. 659, Springer, Berlin, 1978. MR 507448 (80h:26002)
  • [118] David Burdick and F. D. Lesley, Some uniqueness theorems for analytic functions, Amer. Math. Monthly 82 (1975), 152-155. MR 0357766 (50 #10234)
  • [119] Zoltán Buczolich, Solution to the gradient problem of C. E. Weil, Rev. Mat. Iberoamericana 21 (2005), no. 3, 889-910. MR 2231014 (2007c:26012), https://doi.org/10.4171/RMI/439
  • [120] M. C. Calderón-Moreno, Universal functions with small derivatives and extremely fast growth, Analysis (Munich) 22 (2002), no. 1, 57-66. MR 1899914 (2003c:30023)
  • [121] F. S. Cater, Differentiable, nowhere analytic functions, Amer. Math. Monthly 91 (1984), no. 10, 618-624. MR 769526 (86b:26034), https://doi.org/10.2307/2323363
  • [122] Stéphane Charpentier, On the closed subspaces of universal series in Banach spaces and Fréchet spaces, Studia Math. 198 (2010), no. 2, 121-145. MR 2640073 (2011f:30101), https://doi.org/10.4064/sm198-2-2
  • [123] J. Chmielowski, Domains of holomorphy of type $ A^{k}$, Proc. Roy. Irish Acad. Sect. A 80 (1980), no. 1, 97-101. MR 581672 (81m:32014)
  • [124] Charles K. Chui and Milton N. Parnes, Approximation by overconvergence of a power series, J. Math. Anal. Appl. 36 (1971), 693-696. MR 0291472 (45 #563)
  • [125] Krzysztof Ciesielski and Tomasz Natkaniec, Algebraic properties of the class of Sierpiński-Zygmund functions, Topology Appl. 79 (1997), no. 1, 75-99. MR 1462608 (99c:04003), https://doi.org/10.1016/S0166-8641(96)00128-9
  • [126] Krzysztof Ciesielski and Tomasz Natkaniec, On Sierpiński-Zygmund bijections and their inverses, Topology Proc. 22 (1997), no. Spring, 155-164. MR 1657922 (99k:26001)
  • [127] Krzysztof Ciesielski and Janusz Pawlikowski, The covering property axiom, CPA, Cambridge Tracts in Mathematics, vol. 164, Cambridge University Press, Cambridge, 2004. A combinatorial core of the iterated perfect set model. MR 2176267 (2006f:03002)
  • [128] J.A. Conejero, P. Jiménez-Rodríguez, G.A. Muñoz-Fernández, and J.B. Seoane-Sepúlveda, When the Identity Theorem ``seems'' to fail, Amer. Math. Monthly (to appear).
  • [129] Jose A. Conejero, V. Müller, and A. Peris, Hypercyclic behaviour of operators in a hypercyclic $ C_0$-semigroup, J. Funct. Anal. 244 (2007), no. 1, 342-348. MR 2294487 (2007m:47013), https://doi.org/10.1016/j.jfa.2006.12.008
  • [130] Manuel de la Rosa and Charles Read, A hypercyclic operator whose direct sum $ T\oplus T$ is not hypercyclic, J. Operator Theory 61 (2009), no. 2, 369-380. MR 2501011 (2010e:47023)
  • [131] R. Deville and É. Matheron, Infinite games, Banach space geometry and the eikonal equation, Proc. Lond. Math. Soc. (3) 95 (2007), no. 1, 49-68. MR 2329548 (2008f:91007), https://doi.org/10.1112/plms/pdm005
  • [132] E. Diamantopoulos, Ch. Kariofillis, and Ch. Mouratides, Universal Laurent series in finitely connected domains, Arch. Math. (Basel) 91 (2008), no. 2, 145-154. MR 2430798 (2009m:30002), https://doi.org/10.1007/s00013-008-2470-x
  • [133] Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004 (85i:46020)
  • [134] Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London Ltd., London, 1999. MR 1705327 (2001a:46043)
  • [135] Vladimir Drobot and Michał Morayne, Continuous functions with a dense set of proper local maxima, Amer. Math. Monthly 92 (1985), no. 3, 209-211. MR 786345 (86e:26003), https://doi.org/10.2307/2322877
  • [136] A. Dvoretzky and C. A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci. U. S. A. 36 (1950), 192-197. MR 0033975 (11,525a)
  • [137] P.H. Enflo, V.I. Gurariy, and J.B. Seoane-Sepúlveda, Some results and open questions on spaceability in function spaces, Trans. Amer. Math. Soc. (to appear).
  • [138] P.H. Enflo, V.I. Gurariy, and J.B. Seoane-Sepúlveda, On Montgomery's conjecture and the distribution of Dirichlet sums, Preprint (2012).
  • [139] M. Fabián, D. Preiss, J. H. M. Whitfield, and V. E. Zizler, Separating polynomials on Banach spaces, Quart. J. Math. Oxford Ser. (2) 40 (1989), no. 160, 409-422. MR 1033216 (91f:46021), https://doi.org/10.1093/qmath/40.4.409
  • [140] Maite Fernández-Unzueta, Zeroes of polynomials on $ l_\infty $, J. Math. Anal. Appl. 324 (2006), no. 2, 1115-1124. MR 2266546 (2007g:46068), https://doi.org/10.1016/j.jmaa.2006.01.021
  • [141] Jesús Ferrer, Zeroes of real polynomials on $ C(K)$ spaces, J. Math. Anal. Appl. 336 (2007), no. 2, 788-796. MR 2352980 (2008m:46092), https://doi.org/10.1016/j.jmaa.2007.02.083
  • [142] Jesús Ferrer, On the zero-set of real polynomials in non-separable Banach spaces, Publ. Res. Inst. Math. Sci. 43 (2007), no. 3, 685-697. MR 2361792 (2008h:46019)
  • [143] Jesús Ferrer, A note on zeroes of real polynomials in $ C(K)$ spaces, Proc. Amer. Math. Soc. 137 (2009), no. 2, 573-577. MR 2448577 (2010a:46105), https://doi.org/10.1090/S0002-9939-08-09574-9
  • [144] V. P. Fonf, V. I. Gurariy, and M. I. Kadets, An infinite dimensional subspace of $ C[0,1]$ consisting of nowhere differentiable functions, C. R. Acad. Bulgare Sci. 52 (1999), no. 11-12, 13-16. MR 1738120 (2000j:26006)
  • [145] 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)
  • [146] Dieter Gaier, Lectures on complex approximation, Birkhäuser Boston Inc., Boston, MA, 1987. Translated from the German by Renate McLaughlin. MR 894920 (88i:30059b)
  • [147] E.A. Gallardo-Gutiérrez and A. Montes-Rodríguez, The role of the spectrum in the cyclic behavior of composition operators, Memoirs Amer. Math. Soc. 791, AMS, Providence, Rhode Island, 2004.
  • [148] 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 (2012e:26005)
  • [149] 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 (2011g:46042), https://doi.org/10.1090/S0002-9939-2010-10420-3
  • [150] José L. Gámez-Merino, Gustavo A. Muñoz-Fernández, and Juan B. Seoane-Sepúlveda, Lineability and additivity in $ \mathbb{R}^{\mathbb{R}}$, J. Math. Anal. Appl. 369 (2010), no. 1, 265-272. MR 2643865 (2011h:46011), https://doi.org/10.1016/j.jmaa.2010.03.036
  • [151] 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 (2012c:26008), https://doi.org/10.4169/amer.math.monthly.118.02.167
  • [152] José L. Gámez-Merino and Juan B. Seoane-Sepúlveda, An undecidable case of lineability in $ \mathbb{R}^{\mathbb{R}}$, J. Math. Anal. Appl. 401 (2013), no. 2, 959-962. MR 3018041, https://doi.org/10.1016/j.jmaa.2012.10.067
  • [153] Francisco J. García-Pacheco, Vector subspaces of the set of non-norm-attaining functionals, Bull. Aust. Math. Soc. 77 (2008), no. 3, 425-432. MR 2454973 (2009i:46026), https://doi.org/10.1017/S0004972708000348
  • [154] 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 (2011e:46041), https://doi.org/10.1002/mana.200610833
  • [155] 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 (2010h:46072)
  • [156] F. J. García-Pacheco, N. Palmberg, and J. B. Seoane-Sepúlveda, Lineability and algebrability of pathological phenomena in analysis, J. Math. Anal. Appl. 326 (2007), no. 2, 929-939. MR 2280953 (2007i:26003), https://doi.org/10.1016/j.jmaa.2006.03.025
  • [157] F. J. García-Pacheco, C. Pérez-Eslava, and J. B. Seoane-Sepúlveda, Moduleability, algebraic structures, and nonlinear properties, J. Math. Anal. Appl. 370 (2010), no. 1, 159-167. MR 2651137 (2011h:46012), https://doi.org/10.1016/j.jmaa.2010.05.016
  • [158] Francisco Javier García-Pacheco and Daniele Puglisi, Lineability of functionals and operators, Studia Math. 201 (2010), no. 1, 37-47. MR 2733273 (2011j:46010), https://doi.org/10.4064/sm201-1-3
  • [159] F. J. García-Pacheco, F. Rambla-Barreno, and J. B. Seoane-Sepúlveda, $ {\bf Q}$-linear functions, functions with dense graph, and everywhere surjectivity, Math. Scand. 102 (2008), no. 1, 156-160. MR 2420685 (2009c:46002)
  • [160] Francisco J. García-Pacheco and Juan B. Seoane-Sepúlveda, Vector spaces of non-measurable functions, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 6, 1805-1808. MR 2262440 (2007i:28006), https://doi.org/10.1007/s10114-005-0726-y
  • [161] Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in analysis, The Mathesis Series, Holden-Day Inc., San Francisco, Calif., 1964. MR 0169961 (30 #204)
  • [162] Bernard R. Gelbaum and John M. H. Olmsted, Theorems and counterexamples in mathematics, Problem Books in Mathematics, Springer-Verlag, New York, 1990. MR 1066872 (92b:00003)
  • [163] 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
  • [164] S. Głab, P. Kaufmann, and L. Pellegrini, Large structures made of nowhere $ L^p$ functions, arXiv:1207.3818v1 [math.FA].
  • [165] Szymon Głab, Pedro L. Kaufmann, and Leonardo Pellegrini, Spaceability and algebrability of sets of nowhere integrable functions, Proc. Amer. Math. Soc. 141 (2013), no. 6, 2025-2037. MR 3034428, https://doi.org/10.1090/S0002-9939-2012-11574-6
  • [166] J. Glovebnik, The range of vector-valued analytic functions, Ark. Mat. 14 (1976), 113-118.
  • [167] Gilles Godefroy and Joel H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229-269. MR 1111569 (92d:47029), https://doi.org/10.1016/0022-1236(91)90078-J
  • [168] M. Goliński, Invariant subspace problem for classical spaces of functions, J. Funct. Anal. 262 (2012), no. 3, 1251-1273.
  • [169] Manuel González, Fernando León-Saavedra, and Alfonso Montes-Rodríguez, Semi-Fredholm theory: hypercyclic and supercyclic subspaces, Proc. London Math. Soc. (3) 81 (2000), no. 1, 169-189. MR 1757050 (2001g:47013), https://doi.org/10.1112/S0024611500012454
  • [170] Karl Grandjot, Über Grenzwerte ganzer transzendenter Funktionen, Math. Ann. 91 (1924), no. 3-4, 316-320 (German). MR 1512196, https://doi.org/10.1007/BF01556086
  • [171] Sophie Grivaux, Construction of operators with prescribed behaviour, Arch. Math. (Basel) 81 (2003), no. 3, 291-299. MR 2013260 (2004g:47011), https://doi.org/10.1007/s00013-003-0544-3
  • [172] Karl-Goswin Große-Erdmann, Holomorphe Monster und universelle Funktionen, Mitt. Math. Sem. Giessen 176 (1987), iv+84 (German). Dissertation, University of Trier, Trier, 1987. MR 877464 (88i:30060)
  • [173] Karl-Goswin Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345-381. MR 1685272 (2000c:47001), https://doi.org/10.1090/S0273-0979-99-00788-0
  • [174] K.-G. Grosse-Erdmann, Recent developments in hypercyclicity, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 97 (2003), no. 2, 273-286 (English, with English and Spanish summaries). MR 2068180 (2005c:47010)
  • [175] Karl-G. Grosse-Erdmann and Raymond Mortini, Universal functions for composition operators with non-automorphic symbol, J. Anal. Math. 107 (2009), 355-376. MR 2496409 (2010c:30073), https://doi.org/10.1007/s11854-009-0013-4
  • [176] Karl-G. Grosse-Erdmann and Alfredo Peris Manguillot, Linear chaos, Universitext, Springer, London, 2011. MR 2919812
  • [177] Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, AMS Chelsea Publishing, Providence, RI, 2009. Reprint of the 1965 original. MR 2568219 (2010j:32001)
  • [178] V.I. Gurariĭ, Subspaces and bases in spaces of continuous functions, Dokl. Akad. Nauk SSSR 167 (1966), 971-973 (Russian).
  • [179] V.I. Gurariĭ, Linear spaces composed of everywhere nondifferentiable functions, C. R. Acad. Bulgare Sci. 44 (1991), no. 5, 13-16 (Russian).
  • [180] Vladimir I. Gurariy and Wolfgang Lusky, Geometry of Müntz spaces and related questions, Lecture Notes in Mathematics, vol. 1870, Springer-Verlag, Berlin, 2005. MR 2190706 (2007g:46027)
  • [181] Vladimir I. Gurariy and Lucas Quarta, On lineability of sets of continuous functions, J. Math. Anal. Appl. 294 (2004), no. 1, 62-72. MR 2059788 (2005c:46026), https://doi.org/10.1016/j.jmaa.2004.01.036
  • [182] T. R. Hamlett, Compact maps, connected maps and continuity, J. London Math. Soc. (2) 10 (1975), 25-26. MR 0365460 (51 #1712)
  • [183] Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558 (93j:14001)
  • [184] 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 (2001b:46026), https://doi.org/10.1090/S0002-9939-00-05595-7
  • [185] Domingo A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99 (1991), no. 1, 179-190. MR 1120920 (92g:47026), https://doi.org/10.1016/0022-1236(91)90058-D
  • [186] D. Herrero, Hypercyclic operators and chaos, J. Operator Theory 28 (1992), 93-103.
  • [187] H. M. Hilden and L. J. Wallen, Some cyclic and non-cyclic vectors of certain operators, Indiana Univ. Math. J. 23 (1973/74), 557-565. MR 0326452 (48 #4796)
  • [188] Roger A. Horn, Editor's Endnotes, Amer. Math. Monthly 107 (2000), no. 10, 968-969. MR 1543790
  • [189] P. Jiménez-Rodríguez, $ c_0$ is isometrically isomorphic to a subspace of Cantor-Lebesgue functions, J. Math. Anal. Appl. (to appear).
  • [190] P. Jiménez-Rodríguez, G. A. Muñoz-Fernández, and J. B. Seoane-Sepúlveda, Non-Lipschitz functions with bounded gradient and related problems, Linear Algebra Appl. 437 (2012), no. 4, 1174-1181. MR 2926163, https://doi.org/10.1016/j.laa.2012.04.010
  • [191] P. Jiménez-Rodríguez, G.A. Muñoz-Fernández, and J.B. Seoane-Sepúlveda, On Weierstrass' Monsters and lineability, Bull. Belg. Math. Soc. Simon Stevin (to appear).
  • [192] 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 0005906 (3,229e)
  • [193] Francis Edmund Jordan, Cardinal numbers connected with adding Darboux-like functions, ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)-West Virginia University. MR 2697132
  • [194] J.P. Kahane, Baire's category theorem and trigonometric series, J. Analyse Math. 80 (2000), no. 1, 143-182.
  • [195] Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons Inc., New York, 1968. MR 0248482 (40 #1734)
  • [196] Y. Katznelson and Karl Stromberg, Everywhere differentiable, nowhere monotone, functions, Amer. Math. Monthly 81 (1974), 349-354. MR 0335701 (49 #481)
  • [197] Ludger Kaup and Burchard Kaup, Holomorphic functions of several variables, de Gruyter Studies in Mathematics, vol. 3, Walter de Gruyter & Co., Berlin, 1983. An introduction to the fundamental theory; With the assistance of Gottfried Barthel; Translated from the German by Michael Bridgland. MR 716497 (85k:32001)
  • [198] 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 (2006g:26003)
  • [199] St. Kierst and D. Szpilrajn, Sur certaines singularités des fonctions analytiques uniformes, Fundamenta Math. 21 (1933), 276-294.
  • [200] Sung S. Kim and Kil H. Kwon, Smooth $ (C^\infty )$ but nowhere analytic functions, Amer. Math. Monthly 107 (2000), no. 3, 264-266. MR 1742128, https://doi.org/10.2307/2589322
  • [201] Derek Kitson and Richard M. Timoney, Operator ranges and spaceability, J. Math. Anal. Appl. 378 (2011), no. 2, 680-686. MR 2773276 (2012b:46018), https://doi.org/10.1016/j.jmaa.2010.12.061
  • [202] Sergiĭ Kolyada and Ľubomír Snoha, Some aspects of topological transitivity--a survey, Iteration theory (ECIT 94) (Opava), Grazer Math. Ber., vol. 334, Karl-Franzens-Univ. Graz, Graz, 1997, pp. 3-35. MR 1644768
  • [203] T.W. Körner, Fourier analysis, Cambridge University Press, Cambridge, 1988.
  • [204] S. Koumandos, V. Nestoridis, Y.-S. Smyrlis, and V. Stefanopoulos, Universal series in $ \bigcap _{p>1}\ell ^p$, Bull. Lond. Math. Soc. 42 (2010), no. 1, 119-129. MR 2586972 (2011b:30139), https://doi.org/10.1112/blms/bdp102
  • [205] Kenneth Kunen, Set theory. An introduction to independence proofs, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1980. MR 597342 (82f:03001)
  • [206] H. Lebesgue, Leçons sur l'intégration et la recherche des fonctions primitives, Gauthier-Willars, 1904.
  • [207] Fernando León-Saavedra and Alfonso Montes-Rodríguez, Linear structure of hypercyclic vectors, J. Funct. Anal. 148 (1997), no. 2, 524-545. MR 1469352 (98h:47028b), https://doi.org/10.1006/jfan.1996.3084
  • [208] Fernando León-Saavedra and Alfonso Montes-Rodríguez, Spectral theory and hypercyclic subspaces, Trans. Amer. Math. Soc. 353 (2001), no. 1, 247-267. MR 1783790 (2001h:47008), https://doi.org/10.1090/S0002-9947-00-02743-4
  • [209] Fernando León-Saavedra and Vladimír Müller, Hypercyclic sequences of operators, Studia Math. 175 (2006), no. 1, 1-18. MR 2261697 (2007g:47012), https://doi.org/10.4064/sm175-1-1
  • [210] M. Lerch, Ueber die Nichtdifferentiirbarkeit bewisser Funktionen, J. Reine Angew. Math. 103 (1888), 126-138.
  • [211] B. Levine and D. Milman, On linear sets in space $ C$ consisting of functions of bounded variation, Comm. Inst. Sci. Math. Méc. Univ. Kharkoff [Zapiski Inst. Mat. Mech.] (4) 16 (1940), 102-105 (Russian, with English summary). MR 0004712 (3,49g)
  • [212] Joram Lindenstrauss, On subspaces of Banach spaces without quasicomplements, Israel J. Math. 6 (1968), 36-38. MR 0240599 (39 #1946)
  • [213] Jerónimo López-Salazar Codes, Vector spaces of entire functions of unbounded type, Proc. Amer. Math. Soc. 139 (2011), no. 4, 1347-1360. MR 2748427 (2011m:46069), https://doi.org/10.1090/S0002-9939-2010-10817-1
  • [214] Jerónimo López-Salazar, Lineability of the set of holomorphic mappings with dense range, Studia Math. 210 (2012), no. 2, 177-188. MR 2980006, https://doi.org/10.4064/sm210-2-5
  • [215] Mary Lillian Lourenço and Neusa Nogas Tocha, Zeros of complex homogeneous polynomials, Linear Multilinear Algebra 55 (2007), no. 5, 463-469. MR 2363547 (2008k:32017), https://doi.org/10.1080/03081080600628273
  • [216] Wolfgang Luh, Approximation analytischer Funktionen durch überkonvergente Potenzreihen und deren Matrix-Transformierten, Mitt. Math. Sem. Giessen Heft 88 (1970), i+56 (German). MR 0280692 (43 #6411)
  • [217] Wolfgang Luh, Holomorphic monsters, J. Approx. Theory 53 (1988), no. 2, 128-144. MR 945866 (90a:30091), https://doi.org/10.1016/0021-9045(88)90060-3
  • [218] G.R. MacLane, Sequences of derivatives and normal families, J. Anal. Math. 2 (1952), no. 1, 72-87.
  • [219] F. Martínez-Giménez, P. Oprocha, and A. Peris, Distributional chaos for operators with full scrambled sets, Math. Z. (to appear).
  • [220] Quentin Menet, Sous-espaces fermés de séries universelles sur un espace de Fréchet, Studia Math. 207 (2011), no. 2, 181-195 (French, with English summary). MR 2864388, https://doi.org/10.4064/sm207-2-5
  • [221] Q. Menet, Hypercyclic subspaces on Fréchet spaces without continuous norm, arXiv:1302.6447v1 [math.DS], Preprint (2013).
  • [222] Q. Menet, Hypercyclic subspaces and weighted shifts, arXiv:1208.4963v1 [math.DS], Preprint (2013).
  • [223] Alfonso Montes-Rodríguez, Banach spaces of hypercyclic vectors, Michigan Math. J. 43 (1996), no. 3, 419-436. MR 1420585 (98g:47027), https://doi.org/10.1307/mmj/1029005536
  • [224] Alfonso Montes-Rodríguez, A Birkhoff theorem for Riemann surfaces, Rocky Mountain J. Math. 28 (1998), no. 2, 663-693. MR 1651593 (99h:30047), https://doi.org/10.1216/rmjm/1181071794
  • [225] Alfonso Montes-Rodríguez and Héctor N. Salas, Supercyclic subspaces: spectral theory and weighted shifts, Adv. Math. 163 (2001), no. 1, 74-134. MR 1867204 (2002i:47010), https://doi.org/10.1006/aima.2001.2001
  • [226] Alfonso Montes-Rodríguez and Héctor N. Salas, Supercyclic subspaces, Bull. London Math. Soc. 35 (2003), no. 6, 721-737. MR 2000019 (2004d:47022), https://doi.org/10.1112/S002460930300242X
  • [227] Jürgen Müller, Continuous functions with universally divergent Fourier series on small subsets of the circle, C. R. Math. Acad. Sci. Paris 348 (2010), no. 21-22, 1155-1158 (English, with English and French summaries). MR 2738918 (2011m:42010), https://doi.org/10.1016/j.crma.2010.10.026
  • [228] J. Müller, V. Vlachou, and A. Yavrian, Overconvergent series of rational functions and universal Laurent series, J. Anal. Math. 104 (2008), 235-245. MR 2403436 (2009e:30007), https://doi.org/10.1007/s11854-008-0023-7
  • [229] Jorge Mujica, Complex analysis in Banach spaces, North-Holland Mathematics Studies, vol. 120, North-Holland Publishing Co., Amsterdam, 1986. Holomorphic functions and domains of holomorphy in finite and infinite dimensions; Notas de Matemática [Mathematical Notes], 107. MR 842435 (88d:46084)
  • [230] Jorge Mujica, Complex analysis in Banach spaces, North-Holland Mathematics Studies, vol. 120, North-Holland Publishing Co., Amsterdam, 1986. Holomorphic functions and domains of holomorphy in finite and infinite dimensions; Notas de Matemática [Mathematical Notes], 107. MR 842435 (88d:46084)
  • [231] 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), https://doi.org/10.1016/j.laa.2008.01.008
  • [232] T. Natkaniec, Almost continuity, Real Anal. Exchange 17 (1991/92), no. 2, 462-520. MR 1171393 (93e:54009)
  • [233] Tomasz Natkaniec, New cardinal invariants in real analysis, Bull. Polish Acad. Sci. Math. 44 (1996), no. 2, 251-256. MR 1416428 (98a:26007)
  • [234] Vassili Nestoridis, Universal Taylor series, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 5, 1293-1306 (English, with English and French summaries). MR 1427126 (97k:30001)
  • [235] John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York, 1980. A survey of the analogies between topological and measure spaces. MR 584443 (81j:28003)
  • [236] Daniel Pellegrino and Eduardo V. Teixeira, Norm optimization problem for linear operators in classical Banach spaces, Bull. Braz. Math. Soc. (N.S.) 40 (2009), no. 3, 417-431. MR 2540517 (2010i:47066), https://doi.org/10.1007/s00574-009-0019-7
  • [237] Henrik Petersson, Hypercyclic subspaces for Fréchet space operators, J. Math. Anal. Appl. 319 (2006), no. 2, 764-782. MR 2227937 (2007b:47019), https://doi.org/10.1016/j.jmaa.2005.06.042
  • [238] Anatolij Plichko and Andriy Zagorodnyuk, On automatic continuity and three problems of The Scottish book concerning the boundedness of polynomial functionals, J. Math. Anal. Appl. 220 (1998), no. 2, 477-494. MR 1614947 (99c:46050), https://doi.org/10.1006/jmaa.1997.5826
  • [239] Krzysztof Płotka, Sum of Sierpiński-Zygmund and Darboux like functions, Topology Appl. 122 (2002), no. 3, 547-564. MR 1911699 (2003e:26002), https://doi.org/10.1016/S0166-8641(01)00184-5
  • [240] E. E. Posey and J. E. Vaughan, Functions with a proper local maximum in each interval, Amer. Math. Monthly 90 (1983), no. 4, 281-282. MR 700268 (85a:26012), https://doi.org/10.2307/2975762
  • [241] Alfred Pringsheim, Ueber die Multiplication bedingt convergenter Reihen, Math. Ann. 21 (1883), no. 3, 327-378 (German). MR 1510201, https://doi.org/10.1007/BF01443879
  • [242] 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), https://doi.org/10.1016/j.jmaa.2007.05.029
  • [243] C. J. Read, The invariant subspace problem for a class of Banach spaces. II. Hypercyclic operators, Israel J. Math. 63 (1988), no. 1, 1-40. MR 959046 (90b:47013), https://doi.org/10.1007/BF02765019
  • [244] David A. Redett, Strongly annular functions in Bergman space, Comput. Methods Funct. Theory 7 (2007), no. 2, 429-432. MR 2376682 (2008j:46020)
  • [245] L. Rodríguez-Piazza, Every separable Banach space is isometric to a space of continuous nowhere differentiable functions, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3649-3654. MR 1328375 (96d:46007), https://doi.org/10.2307/2161889
  • [246] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22.
  • [247] Haskell P. Rosenthal, On quasi-complemented subspaces of Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 361-364. MR 0227739 (37 #3323)
  • [248] Walter Rudin, Holomorphic maps of discs into $ F$-spaces, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), Springer, Berlin, 1977, pp. 104-108. Lecture Notes in Math., Vol. 599. MR 0499229 (58 #17142)
  • [249] Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157 (88k:00002)
  • [250] Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Inc., New York, 1991. MR 1157815 (92k:46001)
  • [251] Héctor N. Salas, Supercyclicity and weighted shifts, Studia Math. 135 (1999), no. 1, 55-74. MR 1686371 (2000b:47020)
  • [252] Helmut Salzmann and Karl Zeller, Singularitäten unendlich oft differenzierbarer Funktionen, Math. Z. 62 (1955), 354-367 (German). MR 0071479 (17,134b)
  • [253] Juan B. Seoane-Sepúlveda, Chaos and lineability of pathological phenomena in analysis, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)-Kent State University. MR 2709064
  • [254] Juan B. Seoane-Sepúlveda, Explicit constructions of dense common hypercyclic subspaces, Publ. Res. Inst. Math. Sci. 43 (2007), no. 2, 373-384. MR 2341015 (2008g:47021)
  • [255] Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406 (94k:47049)
  • [256] B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (1994), no. 2, 737-754. MR 1227094 (94k:58091), https://doi.org/10.2307/2154504
  • [257] 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)
  • [258] Stanislav Shkarin, On the set of hypercyclic vectors for the differentiation operator, Israel J. Math. 180 (2010), 271-283. MR 2735066 (2012b:47022), https://doi.org/10.1007/s11856-010-0104-z
  • [259] S. Shkarin, Hypercyclic operators on topological vector spaces, J. London Math. Soc. (2) 86 (2012), no. 1, 195-213.
  • [260] Józef Siciak, Highly noncontinuable functions on polynomially convex sets, Complex analysis (Toulouse, 1983) Lecture Notes in Math., vol. 1094, Springer, Berlin, 1984, pp. 173-178. MR 773109 (86e:32009), https://doi.org/10.1007/BFb0099161
  • [261] W. Sierpiński and A. Zygmund, Sur une fonction qui est discontinue sur tout ensemble de puissance du continu, Fund. Math. 4 (1923), 316-318.
  • [262] J. Stallings, Fixed point theorems for connectivity maps, Fund. Math. 47 (1959), 249-263. MR 0117710 (22 #8485)
  • [263] Lynn Arthur Steen and J. Arthur Seebach Jr., Counterexamples in topology, 2nd ed., Springer-Verlag, New York, 1978. MR 507446 (80a:54001)
  • [264] J. Thim, Continuous nowhere differentiable functions, Luleå University of Technology, 2003. Masters Thesis.
  • [265] Stevo Todorčević, Partitioning pairs of countable ordinals, Acta Math. 159 (1987), no. 3-4, 261-294. MR 908147 (88i:04002), https://doi.org/10.1007/BF02392561
  • [266] M. Valdivia, The space $ {\mathcal H}(\Omega ,(z_j))$ of holomorphic functions, J. Math. Anal. Appl. 337 (2008), no. 2, 821-839.
  • [267] M. Valdivia, Spaces of holomorphic functions in regular domains, J. Math. Anal. Appl. 350 (2009), no. 2, 651-662.
  • [268] Daniel J. Velleman, Characterizing continuity, Amer. Math. Monthly 104 (1997), no. 4, 318-322. MR 1450665 (98e:26002), https://doi.org/10.2307/2974580
  • [269] Clifford E. Weil, On properties of derivatives, Trans. Amer. Math. Soc. 114 (1965), 363-376. MR 0176007 (31 #283)
  • [270] Jochen Wengenroth, Hypercyclic operators on non-locally convex spaces, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1759-1761 (electronic). MR 1955262 (2003j:47007), https://doi.org/10.1090/S0002-9939-03-07003-5
  • [271] D. J. White, Functions preserving compactness and connectedness are continuous, J. London Math. Soc. 43 (1968), 714-716. MR 0229211 (37 #4785)
  • [272] A. Wilansky, Semi-Fredholm maps in FK spaces, Math. Z. 144 (1975), 9-12.
  • [273] Gary L. Wise and Eric B. Hall, Counterexamples in probability and real analysis, The Clarendon Press Oxford University Press, New York, 1993. MR 1256489 (95c:60002)
  • [274] Zygmunt Zahorski, Supplément au mémoire ``Sur l'ensemble des points singuliers d'une fonction d'une variable réelle admettant les dérivées de tous les order'', Fund. Math. 36 (1949), 319-320 (French). MR 0035329 (11,718a)
  • [275] Lawrence Zalcman, Uniqueness and nonuniqueness for the Radon transform, Bull. London Math. Soc. 14 (1982), no. 3, 241-245. MR 656606 (83h:42020), https://doi.org/10.1112/blms/14.3.241

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2010): 15A03, 46E10, 46E15, 26B05, 28A20, 47A16, 47L05

Retrieve articles in all journals with MSC (2010): 15A03, 46E10, 46E15, 26B05, 28A20, 47A16, 47L05


Additional Information

Luis Bernal-González
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, Avenida Reina Mercedes, Sevilla, 41080, Spain
Email: lbernal@us.es

Daniel Pellegrino
Affiliation: Departamento de Matemática, Universidade Federal da Paraíba, 58.051-900 - João Pessoa, Brazil
Email: pellegrino@pq.cnpq.br; dmpellegrino@gmail.com

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

DOI: https://doi.org/10.1090/S0273-0979-2013-01421-6
Keywords: Lineability, spaceability, algebrability, real and complex analysis, special functions, operator theory, Baire category theorem, hypercyclic manifolds, zeros of polynomials
Received by editor(s): November 26, 2012
Received by editor(s) in revised form: April 2, 2013
Published electronically: July 15, 2013
Additional Notes: The first author was partially supported by Ministerio de Economía y Competitividad Grant MTM2012-34847-C02-01
The second author was supported by INCT-Matemática, CAPES-NF, CNPq Grants 301237/2009-3 and 477124/2012-7
The third author was supported by MTM2012-34341
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society