Linear subsets of nonlinear sets in topological vector spaces

Bernal-González, Luis; Pellegrino, Daniel; Seoane-Sepúlveda, Juan

doi:10.1090/S0273-0979-2013-01421-6

Linear subsets of nonlinear sets in topological vector spaces
HTML articles powered by AMS MathViewer

by Luis Bernal-González, Daniel Pellegrino and Juan B. Seoane-Sepúlveda PDF

Bull. Amer. Math. Soc. 51 (2014), 71-130 Request permission

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

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
A. Aizpuru, C. Pérez-Eslava, F. J. García-Pacheco, and J. B. Seoane-Sepúlveda, Lineability and coneability of discontinuous functions on $\Bbb R$, Publ. Math. Debrecen 72 (2008), no. 1-2, 129–139. MR 2376864
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, DOI 10.1016/j.laa.2006.02.041
I. Akbarbaglu and S. Maghsoudi, Large structures in certain subsets of Orlicz spaces, Linear Algebra Appl. 438 (2013), no. 11, 4363–4373. MR 3034537, DOI 10.1016/j.laa.2013.01.038
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.
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.
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, DOI 10.1016/j.jmaa.2010.06.059
Shamim I. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), no. 2, 384–390. MR 1469346, DOI 10.1006/jfan.1996.3093
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, DOI 10.2307/2975138
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
D. H. Armitage, Entire functions decaying rapidly on strips, Quaest. Math. 23 (2000), no. 4, 417–424. MR 1810291, DOI 10.2989/16073600009485988
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
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, DOI 10.1142/9789812708441_{0}001
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
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, DOI 10.1023/A:1026278115574
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, DOI 10.1090/conm/435/08365
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, DOI 10.1007/s00020-007-1490-4
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, Raquel Gonzalo, and Andriy Zagorodnyuk, Zeros of real polynomials, Linear and Multilinear Algebra 48 (2000), no. 2, 107–115. MR 1813438, DOI 10.1080/03081080008818662
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 and Petr Hájek, Odd degree polynomials on real Banach spaces, Positivity 11 (2007), no. 1, 143–153. MR 2297328, DOI 10.1007/s11117-006-2035-9
R. M. Aron and P. Hájek, Zero sets of polynomials in several variables, Arch. Math. (Basel) 86 (2006), no. 6, 561–568. MR 2241604, DOI 10.1007/s00013-006-1314-9
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
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
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
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, DOI 10.1016/j.jmaa.2007.08.020
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, DOI 10.1016/j.jmaa.2008.12.033
A.G. Bacharoglu, Universal Taylor series on doubly connected domains, Results in Maths. 53 (2009), no. 1-2, 9–18.
M. Balcerzak, A. Bartoszewicz, and M. Filipczac, Nonseparable spaceability and strong algebrability of sets of continuous singular functions, J. Math. Anal. Appl. (to appear).
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
T. Banakh, A. Bartoszewicz, S. Gła̧b, and E. Szymonik, Algebrability, lineability and the subsums of series, Colloq. Math. 129 (2012), no. 3, 75–85.
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, DOI 10.4064/cm100-1-13
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
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, DOI 10.1090/S0002-9939-2012-11466-2
Artur Bartoszewicz, Marek Bienias, and Szymon Gła̧b, Independent Bernstein sets and algebraic constructions, J. Math. Anal. Appl. 393 (2012), no. 1, 138–143. MR 2921655, DOI 10.1016/j.jmaa.2012.03.007
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 and Szymon Gła̧b, Strong algebrability of sets of sequences and functions, Proc. Amer. Math. Soc. 141 (2013), no. 3, 827–835. MR 3003676, DOI 10.1090/S0002-9939-2012-11377-2
A. Bartoszewicz, S. Gła̧b, D. Pellegrino, and J.B. Seoane-Sepúlveda, Algebrability, nonlinear properties, and special functions, Proc. Amer. Math. Soc., in press.
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
Françoise Bastin, Céline Esser, and Samuel Nicolay, Prevalence of “nowhere analyticity”, Studia Math. 210 (2012), no. 3, 239–246. MR 2983462, DOI 10.4064/sm210-3-4
Frédéric Bayart, Linearity of sets of strange functions, Michigan Math. J. 53 (2005), no. 2, 291–303. MR 2152701, DOI 10.1307/mmj/1123090769
Frédéric Bayart, Topological and algebraic genericity of divergence and universality, Studia Math. 167 (2005), no. 2, 161–181. MR 2134382, DOI 10.4064/sm167-2-4
Frédéric Bayart, Common hypercyclic subspaces, Integral Equations Operator Theory 53 (2005), no. 4, 467–476. MR 2187432, DOI 10.1007/s00020-004-1316-6
Frédéric Bayart, Porosity and hypercyclic operators, Proc. Amer. Math. Soc. 133 (2005), no. 11, 3309–3316. MR 2161154, DOI 10.1090/S0002-9939-05-07842-1
Frédéric Bayart and Sophie Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358 (2006), no. 11, 5083–5117. MR 2231886, DOI 10.1090/S0002-9947-06-04019-0
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, DOI 10.1112/plms/pdl013
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, DOI 10.1112/plms/pdm043
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, DOI 10.14321/realanalexch.29.2.0557
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, DOI 10.1016/j.jfa.2007.05.001
Frédéric Bayart and Étienne Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, vol. 179, Cambridge University Press, Cambridge, 2009. MR 2533318, DOI 10.1017/CBO9780511581113
Frédéric Bayart and Vassili Nestoridis, Universal Taylor series have a strong form of universality, J. Anal. Math. 104 (2008), 69–82. MR 2403430, DOI 10.1007/s11854-008-0017-5
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
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
Bernard Beauzamy, Introduction to operator theory and invariant subspaces, North-Holland Mathematical Library, vol. 42, North-Holland Publishing Co., Amsterdam, 1988. MR 967989
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, DOI 10.1016/j.jmaa.2010.06.011
Luis Bernal-González, A lot of “counterexamples” to Liouville’s theorem, J. Math. Anal. Appl. 201 (1996), no. 3, 1002–1009. MR 1400577, DOI 10.1006/jmaa.1996.0298
Luis Bernal-González, Small entire functions with extremely fast growth, J. Math. Anal. Appl. 207 (1997), no. 2, 541–548. MR 1438930, DOI 10.1006/jmaa.1997.5312
Luis Bernal-González, On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1003–1010. MR 1476119, DOI 10.1090/S0002-9939-99-04657-2
Luis Bernal-González, Densely hereditarily hypercyclic sequences and large hypercyclic manifolds, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3279–3285. MR 1646318, DOI 10.1090/S0002-9939-99-05185-0
Luis Bernal-González, Hypercyclic sequences of differential and antidifferential operators, J. Approx. Theory 96 (1999), no. 2, 323–337. MR 1671201, DOI 10.1006/jath.1998.3237
Luis Bernal-González, Universal images of universal elements, Studia Math. 138 (2000), no. 3, 241–250. MR 1758857, DOI 10.4064/sm-138-3-241-250
L. Bernal-González, Linear Kierst-Szpilrajn theorems, Studia Math. 166 (2005), no. 1, 55–69. MR 2108318, DOI 10.4064/sm166-1-4
L. Bernal-González, Hypercyclic subspaces in Fréchet spaces, Proc. Amer. Math. Soc. 134 (2006), no. 7, 1955–1961. MR 2215764, DOI 10.1090/S0002-9939-05-08242-0
L. Bernal-González, Linear structure of weighted holomorphic non-extendibility, Bull. Austral. Math. Soc. 73 (2006), no. 3, 335–344. MR 2230644, DOI 10.1017/S0004972700035371
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
L. Bernal-González, Lineability of sets of nowhere analytic functions, J. Math. Anal. Appl. 340 (2008), no. 2, 1284–1295. MR 2390929, DOI 10.1016/j.jmaa.2007.09.048
Luis Bernal-González, Algebraic genericity of strict-order integrability, Studia Math. 199 (2010), no. 3, 279–293. MR 2678906, DOI 10.4064/sm199-3-5
Luis Bernal-González, Lineability of universal divergence of Fourier series, Integral Equations Operator Theory 74 (2012), no. 2, 271–279. MR 2983066, DOI 10.1007/s00020-012-1984-6
Luis Bernal-González and Antonio Bonilla, Families of strongly annular functions: linear structure, Rev. Mat. Complut. 26 (2013), no. 1, 283–297. MR 3016629, DOI 10.1007/s13163-011-0080-9
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, DOI 10.1007/s00365-006-0636-5
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, DOI 10.4171/RMI/582
Luis Bernal-González and María Del Carmen Calderón-Moreno, Dense linear manifolds of monsters, J. Approx. Theory 119 (2002), no. 2, 156–180. MR 1939280, DOI 10.1006/jath.2002.3712
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, DOI 10.1016/j.jmaa.2007.10.014
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, DOI 10.1016/j.jmaa.2009.08.049
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, DOI 10.1016/j.jat.2004.06.003
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, DOI 10.1007/s10474-007-5125-7
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, DOI 10.1007/s00020-007-1547-4
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, DOI 10.1016/j.jat.2011.10.005
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, DOI 10.4064/sm157-1-2
Luis Bernal González and Alfonso Montes-Rodríguez, Universal functions for composition operators, Complex Variables Theory Appl. 27 (1995), no. 1, 47–56. MR 1316270, DOI 10.1080/17476939508814804
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, DOI 10.1006/jath.1995.1086
Luis Bernal-González and Manuel Ordóñez Cabrera, Spaceability of strict order integrability, J. Math. Anal. Appl. 385 (2012), no. 1, 303–309. MR 2834257, DOI 10.1016/j.jmaa.2011.06.043
L. Bernal-González and J. A. Prado-Tendero, U-operators, J. Aust. Math. Soc. 78 (2005), no. 1, 59–89. MR 2129489, DOI 10.1017/S1446788700015561
N. Bernardes, A. Bonilla, V. Müller, and A. Peris, Li-Yorke chaos in Linear Dynamics, Preprint (2012).
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, DOI 10.1090/S0002-9939-99-04720-6
J.P. Bès, Dynamics of composition operators with holomorphic symbol, Rev. Real Acad. Cien. Ser. A Mat., in press.
Juan Bès and José A. Conejero, Hypercyclic subspaces in omega, J. Math. Anal. Appl. 316 (2006), no. 1, 16–23. MR 2201746, DOI 10.1016/j.jmaa.2005.04.083
Juan Bès and Alfredo Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), no. 1, 94–112. MR 1710637, DOI 10.1006/jfan.1999.3437
C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 115069, DOI 10.4064/sm-17-2-151-164
G.D. Birkhoff, Démonstration d’un théorème élémentaire sur les fonctions entières, C. R. Acad. Sci. Paris 189 (1929), 473–475.
Errett Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67 (1961), 97–98. MR 123174, DOI 10.1090/S0002-9904-1961-10514-4
H. Blumberg, New properties of all real functions, Trans. Amer. Math. Soc. 82 (1922), 53–61.
P. du Bois-Reymond, Über den Gültigkeitsbereich der Taylorschen Reihenentwicklung, Math. Ann. 21 (1876), 109–117.
D. D. Bonar and F. W. Carroll, Annular functions form a residual set, J. Reine Angew. Math. 272 (1975), 23–24. MR 417428
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, DOI 10.1142/S0218127403007497
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, DOI 10.1016/j.jmaa.2004.03.073
José Bonet and Alfredo Peris, Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal. 159 (1998), no. 2, 587–595. MR 1658096, DOI 10.1006/jfan.1998.3315
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, DOI 10.4064/cm-83-2-155-160
A. Bonilla, Small entire functions with infinite growth index, J. Math. Anal. Appl. 267 (2002), no. 1, 400–404. MR 1886838, DOI 10.1006/jmaa.2001.7767
A. Bonilla, Universal harmonic functions, Quaest. Math. 25 (2002), 527–530.
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, DOI 10.1007/s00020-006-1423-7
A. Bonilla and K.-G. Grosse-Erdmann, Frequently hypercyclic operators and vectors, Ergodic Theory Dynam. Systems 27 (2007), no. 2, 383–404. MR 2308137, DOI 10.1017/S014338570600085X
A. Bonilla and K.-G. Grosse-Erdmann, Frequently hypercyclic subspaces, Monatsh. Math. 168 (2012), no. 3-4, 305–320. MR 2993952, DOI 10.1007/s00605-011-0369-2
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, DOI 10.1017/S0004972700035322
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, DOI 10.1016/j.laa.2012.06.043
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).
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, DOI 10.1016/j.laa.2010.11.012
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
Geraldo Botelho, Daniel Pellegrino, and Pilar Rueda, Cotype and absolutely summing linear operators, Math. Z. 267 (2011), no. 1-2, 1–7. MR 2772238, DOI 10.1007/s00209-009-0591-y
Paul S. Bourdon, Invariant manifolds of hypercyclic vectors, Proc. Amer. Math. Soc. 118 (1993), no. 3, 845–847. MR 1148021, DOI 10.1090/S0002-9939-1993-1148021-4
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.
J. Bourgain, On the distribution of Dirichlet sums, J. Anal. Math. 60 (1993), 21–32.
Andrew M. Bruckner, Differentiation of real functions, Lecture Notes in Mathematics, vol. 659, Springer, Berlin, 1978. MR 507448
David Burdick and F. D. Lesley, Some uniqueness theorems for analytic functions, Amer. Math. Monthly 82 (1975), 152–155. MR 357766, DOI 10.2307/2319660
Zoltán Buczolich, Solution to the gradient problem of C. E. Weil, Rev. Mat. Iberoamericana 21 (2005), no. 3, 889–910. MR 2231014, DOI 10.4171/RMI/439
M. C. Calderón-Moreno, Universal functions with small derivatives and extremely fast growth, Analysis (Munich) 22 (2002), no. 1, 57–66. MR 1899914, DOI 10.1524/anly.2002.22.1.57
F. S. Cater, Differentiable, nowhere analytic functions, Amer. Math. Monthly 91 (1984), no. 10, 618–624. MR 769526, DOI 10.2307/2323363
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, DOI 10.4064/sm198-2-2
J. Chmielowski, Domains of holomorphy of type $A^{k}$, Proc. Roy. Irish Acad. Sect. A 80 (1980), no. 1, 97–101. MR 581672
Charles K. Chui and Milton N. Parnes, Approximation by overconvergence of a power series, J. Math. Anal. Appl. 36 (1971), 693–696. MR 291472, DOI 10.1016/0022-247X(71)90049-7
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
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, DOI 10.1017/CBO9780511546457
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).
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, DOI 10.1016/j.jfa.2006.12.008
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
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, DOI 10.1112/plms/pdm005
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, DOI 10.1007/s00013-008-2470-x
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. MR 1705327, DOI 10.1007/978-1-4471-0869-6
Vladimir Drobot and MichałMorayne, Continuous functions with a dense set of proper local maxima, Amer. Math. Monthly 92 (1985), no. 3, 209–211. MR 786345, DOI 10.2307/2322877
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 33975, DOI 10.1073/pnas.36.3.192
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).
P.H. Enflo, V.I. Gurariy, and J.B. Seoane-Sepúlveda, On Montgomery’s conjecture and the distribution of Dirichlet sums, Preprint (2012).
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, DOI 10.1093/qmath/40.4.409
Maite Fernández-Unzueta, Zeroes of polynomials on $l_\infty$, J. Math. Anal. Appl. 324 (2006), no. 2, 1115–1124. MR 2266546, DOI 10.1016/j.jmaa.2006.01.021
Jesús Ferrer, Zeroes of real polynomials on $C(K)$ spaces, J. Math. Anal. Appl. 336 (2007), no. 2, 788–796. MR 2352980, DOI 10.1016/j.jmaa.2007.02.083
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
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, DOI 10.1090/S0002-9939-08-09574-9
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
James Foran, Fundamentals of real analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 144, Marcel Dekker, Inc., New York, 1991. MR 1201817
Dieter Gaier, Lectures on complex approximation, Birkhäuser Boston, Inc., Boston, MA, 1987. Translated from the German by Renate McLaughlin. MR 894920, DOI 10.1007/978-1-4612-4814-9
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.
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, 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
José L. Gámez-Merino and Juan B. Seoane-Sepúlveda, An undecidable case of lineability in $\Bbb {R}^{\Bbb {R}}$, J. Math. Anal. Appl. 401 (2013), no. 2, 959–962. MR 3018041, DOI 10.1016/j.jmaa.2012.10.067
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, DOI 10.1017/S0004972708000348
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
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, DOI 10.1016/j.jmaa.2006.03.025
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, DOI 10.1016/j.jmaa.2010.05.016
Francisco Javier García-Pacheco and Daniele Puglisi, Lineability of functionals and operators, Studia Math. 201 (2010), no. 1, 37–47. MR 2733273, DOI 10.4064/sm201-1-3
F. J. García-Pacheco, F. Rambla-Barreno, and J. B. Seoane-Sepúlveda, $\textbf {Q}$-linear functions, functions with dense graph, and everywhere surjectivity, Math. Scand. 102 (2008), no. 1, 156–160. MR 2420685, DOI 10.7146/math.scand.a-15057
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, DOI 10.1007/s10114-005-0726-y
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
Bernard R. Gelbaum and John M. H. Olmsted, Theorems and counterexamples in mathematics, Problem Books in Mathematics, Springer-Verlag, New York, 1990. MR 1066872, DOI 10.1007/978-1-4612-0993-5
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
S. Gła̧b, P. Kaufmann, and L. Pellegrini, Large structures made of nowhere $L^p$ functions, arXiv:1207.3818v1 [math.FA].
Szymon Gła̧b, 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, DOI 10.1090/S0002-9939-2012-11574-6
J. Glovebnik, The range of vector-valued analytic functions, Ark. Mat. 14 (1976), 113–118.
Gilles Godefroy and Joel H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229–269. MR 1111569, DOI 10.1016/0022-1236(91)90078-J
M. Goliński, Invariant subspace problem for classical spaces of functions, J. Funct. Anal. 262 (2012), no. 3, 1251–1273.
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, DOI 10.1112/S0024611500012454
Karl Grandjot, Über Grenzwerte ganzer transzendenter Funktionen, Math. Ann. 91 (1924), no. 3-4, 316–320 (German). MR 1512196, DOI 10.1007/BF01556086
Sophie Grivaux, Construction of operators with prescribed behaviour, Arch. Math. (Basel) 81 (2003), no. 3, 291–299. MR 2013260, DOI 10.1007/s00013-003-0544-3
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
Karl-Goswin Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345–381. MR 1685272, DOI 10.1090/S0273-0979-99-00788-0
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
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, DOI 10.1007/s11854-009-0013-4
Karl-G. Grosse-Erdmann and Alfredo Peris Manguillot, Linear chaos, Universitext, Springer, London, 2011. MR 2919812, DOI 10.1007/978-1-4471-2170-1
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, DOI 10.1090/chel/368
V.I. Gurariĭ, Subspaces and bases in spaces of continuous functions, Dokl. Akad. Nauk SSSR 167 (1966), 971–973 (Russian).
V.I. Gurariĭ, Linear spaces composed of everywhere nondifferentiable functions, C. R. Acad. Bulgare Sci. 44 (1991), no. 5, 13–16 (Russian).
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, DOI 10.1007/11551621
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, DOI 10.1016/j.jmaa.2004.01.036
T. R. Hamlett, Compact maps, connected maps and continuity, J. London Math. Soc. (2) 10 (1975), 25–26. MR 365460, DOI 10.1112/jlms/s2-10.1.25
Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558, DOI 10.1007/978-1-4757-2189-8
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
Domingo A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99 (1991), no. 1, 179–190. MR 1120920, DOI 10.1016/0022-1236(91)90058-D
D. Herrero, Hypercyclic operators and chaos, J. Operator Theory 28 (1992), 93–103.
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 326452, DOI 10.1512/iumj.1974.23.23046
Roger A. Horn, Editor’s Endnotes, Amer. Math. Monthly 107 (2000), no. 10, 968–969. MR 1543790
P. Jiménez-Rodríguez, $c_0$ is isometrically isomorphic to a subspace of Cantor–Lebesgue functions, J. Math. Anal. Appl. (to appear).
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, DOI 10.1016/j.laa.2012.04.010
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).
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
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
J.P. Kahane, Baire’s category theorem and trigonometric series, J. Analyse Math. 80 (2000), no. 1, 143–182.
Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
Y. Katznelson and Karl Stromberg, Everywhere differentiable, nowhere monotone, functions, Amer. Math. Monthly 81 (1974), 349–354. MR 335701, DOI 10.2307/2318996
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, DOI 10.1515/9783110838350
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
St. Kierst and D. Szpilrajn, Sur certaines singularités des fonctions analytiques uniformes, Fundamenta Math. 21 (1933), 276–294.
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, DOI 10.2307/2589322
Derek Kitson and Richard M. Timoney, Operator ranges and spaceability, J. Math. Anal. Appl. 378 (2011), no. 2, 680–686. MR 2773276, DOI 10.1016/j.jmaa.2010.12.061
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
T.W. Körner, Fourier analysis, Cambridge University Press, Cambridge, 1988.
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, DOI 10.1112/blms/bdp102
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
H. Lebesgue, Leçons sur l’intégration et la recherche des fonctions primitives, Gauthier-Willars, 1904.
Fernando León-Saavedra and Alfonso Montes-Rodríguez, Linear structure of hypercyclic vectors, J. Funct. Anal. 148 (1997), no. 2, 524–545. MR 1469352, DOI 10.1006/jfan.1996.3084
Fernando León-Saavedra and Alfonso Montes-Rodríguez, Spectral theory and hypercyclic subspaces, Trans. Amer. Math. Soc. 353 (2001), no. 1, 247–267. MR 1783790, DOI 10.1090/S0002-9947-00-02743-4
Fernando León-Saavedra and Vladimír Müller, Hypercyclic sequences of operators, Studia Math. 175 (2006), no. 1, 1–18. MR 2261697, DOI 10.4064/sm175-1-1
M. Lerch, Ueber die Nichtdifferentiirbarkeit bewisser Funktionen, J. Reine Angew. Math. 103 (1888), 126–138.
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
Joram Lindenstrauss, On subspaces of Banach spaces without quasicomplements, Israel J. Math. 6 (1968), 36–38. MR 240599, DOI 10.1007/BF02771603
Jerónimo López-Salazar Codes, Vector spaces of entire functions of unbounded type, Proc. Amer. Math. Soc. 139 (2011), no. 4, 1347–1360. MR 2748427, DOI 10.1090/S0002-9939-2010-10817-1
Jerónimo López-Salazar, Lineability of the set of holomorphic mappings with dense range, Studia Math. 210 (2012), no. 2, 177–188. MR 2980006, DOI 10.4064/sm210-2-5
Mary Lillian Lourenço and Neusa Nogas Tocha, Zeros of complex homogeneous polynomials, Linear Multilinear Algebra 55 (2007), no. 5, 463–469. MR 2363547, DOI 10.1080/03081080600628273
Wolfgang Luh, Approximation analytischer Funktionen durch überkonvergente Potenzreihen und deren Matrix-Transformierten, Mitt. Math. Sem. Giessen 88 (1970), i+56 (German). MR 280692
Wolfgang Luh, Holomorphic monsters, J. Approx. Theory 53 (1988), no. 2, 128–144. MR 945866, DOI 10.1016/0021-9045(88)90060-3
G.R. MacLane, Sequences of derivatives and normal families, J. Anal. Math. 2 (1952), no. 1, 72–87.
F. Martínez-Giménez, P. Oprocha, and A. Peris, Distributional chaos for operators with full scrambled sets, Math. Z. (to appear).
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, DOI 10.4064/sm207-2-5
Q. Menet, Hypercyclic subspaces on Fréchet spaces without continuous norm, arXiv:1302.6447v1 [math.DS], Preprint (2013).
Q. Menet, Hypercyclic subspaces and weighted shifts, arXiv:1208.4963v1 [math.DS], Preprint (2013).
Alfonso Montes-Rodríguez, Banach spaces of hypercyclic vectors, Michigan Math. J. 43 (1996), no. 3, 419–436. MR 1420585, DOI 10.1307/mmj/1029005536
Alfonso Montes-Rodríguez, A Birkhoff theorem for Riemann surfaces, Rocky Mountain J. Math. 28 (1998), no. 2, 663–693. MR 1651593, DOI 10.1216/rmjm/1181071794
Alfonso Montes-Rodríguez and Héctor N. Salas, Supercyclic subspaces: spectral theory and weighted shifts, Adv. Math. 163 (2001), no. 1, 74–134. MR 1867204, DOI 10.1006/aima.2001.2001
Alfonso Montes-Rodríguez and Héctor N. Salas, Supercyclic subspaces, Bull. London Math. Soc. 35 (2003), no. 6, 721–737. MR 2000019, DOI 10.1112/S002460930300242X
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, DOI 10.1016/j.crma.2010.10.026
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, DOI 10.1007/s11854-008-0023-7
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
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
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
T. Natkaniec, Almost continuity, Real Anal. Exchange 17 (1991/92), no. 2, 462–520. MR 1171393
Tomasz Natkaniec, New cardinal invariants in real analysis, Bull. Polish Acad. Sci. Math. 44 (1996), no. 2, 251–256. MR 1416428
Vassili Nestoridis, Universal Taylor series, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 5, 1293–1306 (English, with English and French summaries). MR 1427126
John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. MR 584443
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, DOI 10.1007/s00574-009-0019-7
Henrik Petersson, Hypercyclic subspaces for Fréchet space operators, J. Math. Anal. Appl. 319 (2006), no. 2, 764–782. MR 2227937, DOI 10.1016/j.jmaa.2005.06.042
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, DOI 10.1006/jmaa.1997.5826
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
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, DOI 10.2307/2975762
Alfred Pringsheim, Ueber die Multiplication bedingt convergenter Reihen, Math. Ann. 21 (1883), no. 3, 327–378 (German). MR 1510201, DOI 10.1007/BF01443879
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, DOI 10.1016/j.jmaa.2007.05.029
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, DOI 10.1007/BF02765019
David A. Redett, Strongly annular functions in Bergman space, Comput. Methods Funct. Theory 7 (2007), no. 2, 429–432. MR 2376682, DOI 10.1007/BF03321655
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, DOI 10.1090/S0002-9939-1995-1328375-8
S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17–22.
Haskell P. Rosenthal, On quasi-complemented subspaces of Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 361–364. MR 227739, DOI 10.1073/pnas.59.2.361
Walter Rudin, Holomorphic maps of discs into $F$-spaces, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976) Lecture Notes in Math., Vol. 599, Springer, Berlin, 1977, pp. 104–108. MR 0499229
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
Héctor N. Salas, Supercyclicity and weighted shifts, Studia Math. 135 (1999), no. 1, 55–74. MR 1686371, DOI 10.4064/sm-135-1-55-74
Helmut Salzmann and Karl Zeller, Singularitäten unendlich oft differenzierbarer Funktionen, Math. Z. 62 (1955), 354–367 (German). MR 71479, DOI 10.1007/BF01180644
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
Juan B. Seoane-Sepúlveda, Explicit constructions of dense common hypercyclic subspaces, Publ. Res. Inst. Math. Sci. 43 (2007), no. 2, 373–384. MR 2341015
Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406, DOI 10.1007/978-1-4612-0887-7
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, DOI 10.1090/S0002-9947-1994-1227094-X
Juichi Shinoda, Some consequences of Martin’s axiom and the negation of the continuum hypothesis, Nagoya Math. J. 49 (1973), 117–125. MR 319754
Stanislav Shkarin, On the set of hypercyclic vectors for the differentiation operator, Israel J. Math. 180 (2010), 271–283. MR 2735066, DOI 10.1007/s11856-010-0104-z
S. Shkarin, Hypercyclic operators on topological vector spaces, J. London Math. Soc. (2) 86 (2012), no. 1, 195–213.
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, DOI 10.1007/BFb0099161
W. Sierpiński and A. Zygmund, Sur une fonction qui est discontinue sur tout ensemble de puissance du continu, Fund. Math. 4 (1923), 316–318.
J. Stallings, Fixed point theorems for connectivity maps, Fund. Math. 47 (1959), 249–263. MR 117710, DOI 10.4064/fm-47-3-249-263
Lynn Arthur Steen and J. Arthur Seebach Jr., Counterexamples in topology, 2nd ed., Springer-Verlag, New York-Heidelberg, 1978. MR 507446
J. Thim, Continuous nowhere differentiable functions, Luleå University of Technology, 2003. Masters Thesis.
Stevo Todorčević, Partitioning pairs of countable ordinals, Acta Math. 159 (1987), no. 3-4, 261–294. MR 908147, DOI 10.1007/BF02392561
M. Valdivia, The space ${\mathcal H}(\Omega ,(z_j))$ of holomorphic functions, J. Math. Anal. Appl. 337 (2008), no. 2, 821–839.
M. Valdivia, Spaces of holomorphic functions in regular domains, J. Math. Anal. Appl. 350 (2009), no. 2, 651–662.
Daniel J. Velleman, Characterizing continuity, Amer. Math. Monthly 104 (1997), no. 4, 318–322. MR 1450665, DOI 10.2307/2974580
Clifford E. Weil, On properties of derivatives, Trans. Amer. Math. Soc. 114 (1965), 363–376. MR 176007, DOI 10.1090/S0002-9947-1965-0176007-2
Jochen Wengenroth, Hypercyclic operators on non-locally convex spaces, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1759–1761. MR 1955262, DOI 10.1090/S0002-9939-03-07003-5
D. J. White, Functions preserving compactness and connectedness are continuous, J. London Math. Soc. 43 (1968), 714–716. MR 229211, DOI 10.1112/jlms/s1-43.1.714
A. Wilansky, Semi-Fredholm maps in FK spaces, Math. Z. 144 (1975), 9–12.
Gary L. Wise and Eric B. Hall, Counterexamples in probability and real analysis, The Clarendon Press, Oxford University Press, New York, 1993. MR 1256489
Zygmunt Zahorski, Supplément au mémoire “Sur l’ensemble des points singuliers d’une fonction d’une variable réelle admettant les dérivées de tous les orders.”, Fund. Math. 36 (1949), 319–320 (French). MR 35329, DOI 10.4064/fm-36-1-319-320
Lawrence Zalcman, Uniqueness and nonuniqueness for the Radon transform, Bull. London Math. Soc. 14 (1982), no. 3, 241–245. MR 656606, DOI 10.1112/blms/14.3.241