## Linear subsets of nonlinear sets in topological vector spaces.(English)Zbl 1292.46004

In the last few decades, increasing attention has been given to the following type of “phenomenon”: We are given a linear space $$X$$ and a (possibly strange) property $$\mathcal P$$ that some elements of $$X$$ might satisfy. Next, we consider the subset $$\mathcal S(\mathcal P) \subset X$$, $$\mathcal S(\mathcal P) \equiv \{x \in X \;| \;x$$ satisfies $$\mathcal P\} \cup \{0\}$$. The motivation for this fascinating survey is the fact that in many cases, for many $$X$$ and many $$\mathcal P$$, the set $$\mathcal S(\mathcal P)$$ contains large linear subspaces. If this occurs, we say that $$\mathcal P$$ is a lineable property.
As an example, let $$X = \mathcal C[0,1]$$ and let $$\mathcal P$$ be the property that a function is continuous but nowhere differentiable. It is known that this is a lineable property. It is also spaceable; that is, the set $$\mathcal S(\mathcal P)$$ contains an infinite-dimensional closed subspace (cf. [V. Fonf et al., C. R. Acad. Bulg. Sci. 52, No. 11–12, 13–16 (1999; Zbl 0945.26010)] and [L. Rodríguez-Piazza, Proc. Am. Math. Soc. 123, No. 12, 3649–3654 (1995; Zbl 0844.46007)]). In fact, even more is true: $$\mathcal S(\mathcal P)$$ is algebrable, essentially meaning that there is an infinitely generated algebra $$\mathcal A \subset \mathcal S(\mathcal P)$$ (cf. [F. Bayart and L. Quarta, Isr. J. Math. 158, 285–296 (2007; Zbl 1138.46017)]). It seems difficult to guess what properties are lineable, spaceable, algebrable, or one of several other natural conditions. For instance, with the same $$X = \mathcal C[0,1]$$ but now $$\mathcal P$$ being the much nicer property that the function is everywhere differentiable, it turns out that $$\mathcal S(\mathcal P)$$ is nothing more than lineable (noting, e.g., that each polynomial is in $$\mathcal S(\mathcal P)$$).
Nearly half of this article deals with such situations, where $$X$$ is a space of continuous, or differentiable, or measurable, or …functions. Another section focuses on the relation between lineability and hypercyclicity. Recall that if $$X$$ is a separable Banach or Fréchet space, an operator $$T:X \to X$$ is called hypercyclic on $$X$$ if there is a vector $$x_0 \in X$$ whose orbit under $$T$$, $$\mathrm{orb}(T,x_0) = \{x_0, T(x_0), T^2(x_0),\dots\}$$ is dense in $$X.$$ In this case, we say that $$x_0$$ is a hypercyclic vector for $$T$$. One basic example, due to S. Rolewicz [Stud. Math. 32, 17–22 (1969; Zbl 0174.44203)], is the weighted backward shift $$2B:\ell_2 \to \ell_2$$, $$2B(x_1,x_2,\dots) = 2(x_2,x_3,\dots)$$. It was observed by P. S. Bourdon [Proc. Am. Math. Soc. 118, No. 3, 845–847 (1993; Zbl 0809.47005)] that if $$T$$ is hypercyclic on $$X$$, then $$\{x \in X \;| \;x$$ is hypercyclic for $$T \} \cup \{0\}$$ contains an infinite-dimensional vector space. In fact, in the particular example of the weighted backward shift, this set does not contain an infinite-dimensional closed subspace. In other words, we have lineability but not spaceability for such a hypercyclic operator. On the other hand, when $$X = \mathcal H(\mathbb C),$$ the space of entire functions, and $$D$$ is the (somewhat) analogous differentiation operator, $$D(f) = f^\prime,$$ then G. R. MacLane [J. Anal. Math. 2, 72–87 (1952; Zbl 0049.05603)] showed that there are indeed functions $$f_0 \in \mathcal H(\mathbb C)$$ whose orbit under $$D$$ is dense in $$\mathcal H(\mathbb C)$$. Moreover, S. Shkarin [Isr. J. Math. 180, 271–283 (2010; Zbl 1218.47017)] proved that there is a closed subspace of such functions $$f_0$$ (assuming, of course, that the $$0$$-function is included).
The article concludes with a discussion of the existence of linear subspaces of $$P^{-1}(0),$$ where $$P:X \to {\mathbb {K}}$$ is a $${\mathbb {K}} = \mathbb R$$- or $$\mathbb C$$-valued polynomial on a finite- or infinite-dimensional space $$X$$ such that $$P(0) = 0$$, followed by an extensive bibliography.
One must be careful to avoid making sweeping generalizations about lineability properties. For instance, building on an example of C. J. Read [“Banach spaces with no proximinal subspaces of codimension 2”, arXiv:1307.7958, Isr. J. Math. 223, 493–504 (2018; Zbl 1397.46010)], very recently M. Rmoutil [Private communication, J. Funct. Anal. 272, No. 3, 918–928 (2017; Zbl 1359.46008)] has solved an important lineability problem due to P. Bandyopadhyay and G. Godefroy [J. Convex Anal. 13, No. 3–4, 489–497 (2006; Zbl 1118.46024)] involving the Bishop-Phelps theorem: There is a renorming of $$c_0$$ such that the set of norm-attaining continuous linear functionals in its dual does not contain a 2-dimensional subspace. In another direction, in very many cases, the set $$\mathcal S(\mathcal P)$$ contains a dense $$G_\delta$$ set, and so it is natural to wonder whether the Baire category theorem is somehow related to lineability. As the authors note, this does not appear to be the case. For instance, using $$X = \mathcal C[0,1]$$, where only $$\mathbb R$$-valued functions are considered [V. I. Gurariy and L. Quarta, J. Math. Anal. Appl. 294, No. 1, 62–72 (2004; Zbl 1053.46014)], let $$\mathcal P = \{f \in X \;| \;f$$ attains its maximum at exactly one point $$\} \cup \{0\}$$. They showed that, although $$\mathcal S(\mathcal P)$$ is a dense $$G_\delta$$ in $$\mathcal C[0,1],$$ it does not even contain a 2-dimensional subspace.
To summarize, this survey is likely to be, and deserves to be, studied for some considerable time.

### MSC:

 46A99 Topological linear spaces and related structures 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 15A03 Vector spaces, linear dependence, rank, lineability 46E10 Topological linear spaces of continuous, differentiable or analytic functions 46E15 Banach spaces of continuous, differentiable or analytic functions 26B05 Continuity and differentiation questions 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 47A16 Cyclic vectors, hypercyclic and chaotic operators 47L05 Linear spaces of operators
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 [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. · Zbl 1144.46013 [2] 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. · Zbl 1164.26312 [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. · Zbl 1106.40001 · doi: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. · Zbl 1292.46017 · doi: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. · Zbl 1308.43005 [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. · Zbl 1287.47007 [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. · Zbl 1202.26009 · doi: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. , https://doi.org/10.1006/jfan.1996.3093 Fernando León-Saavedra and Alfonso Montes-Rodríguez, Linear structure of hypercyclic vectors, J. Funct. Anal. 148 (1997), no. 2, 524 – 545. · Zbl 0999.47009 · doi:10.1006/jfan.1996.3084 [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. · Zbl 0838.30035 · doi: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. · Zbl 0948.31001 [11] D. H. Armitage, Entire functions decaying rapidly on strips, Quaest. Math. 23 (2000), no. 4, 417 – 424. · Zbl 0969.30012 · doi: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). · Zbl 1027.46052 [13] Richard M. Aron, Linearity in non-linear situations, Advanced courses of mathematical analysis. II, World Sci. Publ., Hackensack, NJ, 2007, pp. 1 – 15. · Zbl 1149.46022 · doi: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. · Zbl 1106.47010 [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. · Zbl 1044.46036 · doi: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. · Zbl 1149.46024 · doi: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. · Zbl 1142.47009 · doi: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. · Zbl 1207.46025 [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). · Zbl 1036.46033 [20] 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. · Zbl 1210.26008 · doi: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. · Zbl 0972.12002 · doi:10.1080/03081080008818662 [22] 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. · Zbl 1069.26006 [23] Richard M. Aron and Petr Hájek, Odd degree polynomials on real Banach spaces, Positivity 11 (2007), no. 1, 143 – 153. · Zbl 1158.46030 · doi: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. · Zbl 1106.46029 · doi: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. · Zbl 1102.42001 · doi: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. · Zbl 0918.58007 [27] 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. · Zbl 1130.46013 [28] Antonio Avilés and Stevo Todorcevic, Zero subspaces of polynomials on \?$$_{1}$$(\Gamma ), J. Math. Anal. Appl. 350 (2009), no. 2, 427 – 435. · Zbl 1391.46056 · doi: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. · Zbl 1171.26005 · doi: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. · Zbl 1178.30002 [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). · Zbl 1314.46033 [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. · Zbl 0905.26001 · doi: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. · Zbl 1066.46040 · doi: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. · Zbl 1118.46024 [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. · Zbl 1285.46016 [37] Artur Bartoszewicz, Marek Bienias, and Szymon Głb, Independent Bernstein sets and algebraic constructions, J. Math. Anal. Appl. 393 (2012), no. 1, 138 – 143. · Zbl 1253.26002 · doi:10.1016/j.jmaa.2012.03.007 [38] Artur Bartoszewicz and Szymon Głb, Algebrability of conditionally convergent series with Cauchy product, J. Math. Anal. Appl. 385 (2012), no. 2, 693 – 697. · Zbl 1230.40008 · doi:10.1016/j.jmaa.2011.07.008 [39] Artur Bartoszewicz and Szymon Głb, Strong algebrability of sets of sequences and functions, Proc. Amer. Math. Soc. 141 (2013), no. 3, 827 – 835. · Zbl 1261.15003 [40] A. Bartoszewicz, S. Głab, D. Pellegrino, and J.B. Seoane-Sepúlveda, Algebrability, nonlinear properties, and special functions, Proc. Amer. Math. Soc., in press. · Zbl 1287.46040 [41] Artur Bartoszewicz, Szymon Głb, and Tadeusz Poreda, On algebrability of nonabsolutely convergent series, Linear Algebra Appl. 435 (2011), no. 5, 1025 – 1028. · Zbl 1229.40001 · doi: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. · Zbl 1282.46023 · doi:10.4064/sm210-3-4 [43] Frédéric Bayart, Linearity of sets of strange functions, Michigan Math. J. 53 (2005), no. 2, 291 – 303. · Zbl 1092.46006 · doi:10.1307/mmj/1123090769 [44] Frédéric Bayart, Topological and algebraic genericity of divergence and universality, Studia Math. 167 (2005), no. 2, 161 – 181. · Zbl 1076.46012 · doi:10.4064/sm167-2-4 [45] Frédéric Bayart, Common hypercyclic subspaces, Integral Equations Operator Theory 53 (2005), no. 4, 467 – 476. · Zbl 1120.47003 · doi:10.1007/s00020-004-1316-6 [46] Frédéric Bayart, Porosity and hypercyclic operators, Proc. Amer. Math. Soc. 133 (2005), no. 11, 3309 – 3316. · Zbl 1202.47009 [47] Frédéric Bayart and Sophie Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358 (2006), no. 11, 5083 – 5117. · Zbl 1115.47005 [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. · Zbl 1115.47006 · doi: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. · Zbl 1147.30003 · doi: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. · Zbl 1068.42005 [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. · Zbl 1131.47006 · doi: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. · Zbl 1187.47001 [53] Frédéric Bayart and Vassili Nestoridis, Universal Taylor series have a strong form of universality, J. Anal. Math. 104 (2008), 69 – 82. · Zbl 1156.30003 · doi: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. · Zbl 1138.46017 · doi: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). · Zbl 0612.47003 [56] Bernard Beauzamy, Introduction to operator theory and invariant subspaces, North-Holland Mathematical Library, vol. 42, North-Holland Publishing Co., Amsterdam, 1988. · Zbl 0663.47002 [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. · Zbl 1214.47012 · doi: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. · Zbl 0855.30030 · doi: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. · Zbl 0872.30022 · doi: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. · Zbl 0911.47020 [61] Luis Bernal-González, Densely hereditarily hypercyclic sequences and large hypercyclic manifolds, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3279 – 3285. · Zbl 0933.47002 [62] Luis Bernal-González, Hypercyclic sequences of differential and antidifferential operators, J. Approx. Theory 96 (1999), no. 2, 323 – 337. · Zbl 0957.47009 · doi:10.1006/jath.1998.3237 [63] Luis Bernal-González, Universal images of universal elements, Studia Math. 138 (2000), no. 3, 241 – 250. · Zbl 0959.47002 [64] L. Bernal-González, Linear Kierst-Szpilrajn theorems, Studia Math. 166 (2005), no. 1, 55 – 69. · Zbl 1062.30003 · doi:10.4064/sm166-1-4 [65] L. Bernal-González, Hypercyclic subspaces in Fréchet spaces, Proc. Amer. Math. Soc. 134 (2006), no. 7, 1955 – 1961. · Zbl 1094.47010 [66] L. Bernal-González, Linear structure of weighted holomorphic non-extendibility, Bull. Austral. Math. Soc. 73 (2006), no. 3, 335 – 344. · Zbl 1104.30003 · doi:10.1017/S0004972700035371 [67] L. Bernal-González, Dense-lineability in spaces of continuous functions, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3163 – 3169. · Zbl 1160.46016 [68] L. Bernal-González, Lineability of sets of nowhere analytic functions, J. Math. Anal. Appl. 340 (2008), no. 2, 1284 – 1295. · Zbl 1158.46016 · doi: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. · Zbl 1232.46026 · doi: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. · Zbl 1282.47008 · doi: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. · Zbl 1270.30001 · doi: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 \?-analytic functions along arbitrary curves, Constr. Approx. 25 (2007), no. 2, 211 – 219. · Zbl 1109.30031 · doi: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. · Zbl 1186.30003 · doi:10.4171/RMI/582 [74] Luis Bernal-González and María Del Carmen Calderón-Moreno, Dense linear manifolds of monsters, J. Approx. Theory 119 (2002), no. 2, 156 – 180. · Zbl 1026.47022 · doi: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. · Zbl 1160.30351 · doi: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. · Zbl 1182.30101 · doi: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. · Zbl 1053.30021 · doi: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. · Zbl 1129.47006 · doi: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. · Zbl 1149.47023 · doi: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. · Zbl 1250.30053 · doi: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. · Zbl 1032.47006 · doi: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. · Zbl 0838.30032 [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. · Zbl 0831.30024 · doi: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. · Zbl 1234.46028 · doi: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. · Zbl 1079.30054 · doi:10.1017/S1446788700015561 [86] N. Bernardes, A. Bonilla, V. Müller, and A. Peris, Li-Yorke chaos in Linear Dynamics, Preprint (2012). · Zbl 1352.37100 [87] Juan P. Bès, Invariant manifolds of hypercyclic vectors for the real scalar case, Proc. Amer. Math. Soc. 127 (1999), no. 6, 1801 – 1804. · Zbl 0914.47005 [88] J.P. Bès, Dynamics of composition operators with holomorphic symbol, Rev. Real Acad. Cien. Ser. A Mat., in press. · Zbl 1298.47015 [89] Juan Bès and José A. Conejero, Hypercyclic subspaces in omega, J. Math. Anal. Appl. 316 (2006), no. 1, 16 – 23. · Zbl 1094.47011 · doi: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. · Zbl 0941.47002 · doi: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. · Zbl 0084.09805 [92] G.D. Birkhoff, Démonstration d’un théorème élémentaire sur les fonctions entières, C. R. Acad. Sci. Paris 189 (1929), 473-475. · JFM 55.0192.07 [93] Errett Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67 (1961), 97 – 98. · Zbl 0098.07905 [94] H. Blumberg, New properties of all real functions, Trans. Amer. Math. Soc. 82 (1922), 53-61. · JFM 48.0264.01 [95] P. du Bois-Reymond, Über den Gültigkeitsbereich der Taylorschen Reihenentwicklung, Math. Ann. 21 (1876), 109-117. · JFM 15.0183.01 [96] D. D. Bonar and F. W. Carroll, Annular functions form a residual set, J. Reine Angew. Math. 272 (1975), 23 – 24. · Zbl 0297.30031 [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). · Zbl 1079.47008 · doi: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. · Zbl 1062.47011 · doi: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. · Zbl 0926.47011 · doi: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. · Zbl 0967.31001 [101] A. Bonilla, Small entire functions with infinite growth index, J. Math. Anal. Appl. 267 (2002), no. 1, 400 – 404. · Zbl 1003.30021 · doi:10.1006/jmaa.2001.7767 [102] A. Bonilla, Universal harmonic functions, Quaest. Math. 25 (2002), 527-530. · Zbl 1028.31002 [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. · Zbl 1114.47004 · doi: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. · Zbl 1119.47011 · doi:10.1017/S014338570600085X [105] A. Bonilla and K.-G. Grosse-Erdmann, Frequently hypercyclic subspaces, Monatsh. Math. 168 (2012), no. 3-4, 305 – 320. · Zbl 1290.47007 · doi: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. · Zbl 1121.49015 · doi: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. · Zbl 1258.46007 · doi: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). · Zbl 1284.46022 [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. · Zbl 1226.46017 · doi: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. · Zbl 1167.47016 · doi:10.1016/j.jmaa.2009.03.062 [111] G. Botelho, V. V. Fávaro, D. Pellegrino, and J. B. Seoane-Sepúlveda, \?_{\?}[0,1]_s\?_{\?>\?}\?_{\?}[0,1] is spaceable for every \?>0, Linear Algebra Appl. 436 (2012), no. 9, 2963 – 2965. · Zbl 1246.46016 · doi: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. · Zbl 1230.46039 · doi: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. · Zbl 1226.47017 · doi: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. · Zbl 0809.47005 [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. · Zbl 0793.11020 [117] Andrew M. Bruckner, Differentiation of real functions, Lecture Notes in Mathematics, vol. 659, Springer, Berlin, 1978. · Zbl 0382.26002 [118] David Burdick and F. D. Lesley, Some uniqueness theorems for analytic functions, Amer. Math. Monthly 82 (1975), 152 – 155. · Zbl 0306.30004 · doi:10.2307/2319660 [119] Zoltán Buczolich, Solution to the gradient problem of C. E. Weil, Rev. Mat. Iberoamericana 21 (2005), no. 3, 889 – 910. · Zbl 1116.26007 · doi: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. · Zbl 0999.30022 · doi:10.1524/anly.2002.22.1.57 [121] F. S. Cater, Differentiable, nowhere analytic functions, Amer. Math. Monthly 91 (1984), no. 10, 618 – 624. · Zbl 0598.26035 · doi: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. · Zbl 1201.30005 · doi:10.4064/sm198-2-2 [123] J. Chmielowski, Domains of holomorphy of type \?^{\?}, Proc. Roy. Irish Acad. Sect. A 80 (1980), no. 1, 97 – 101. · Zbl 0421.32013 [124] Charles K. Chui and Milton N. Parnes, Approximation by overconvergence of a power series, J. Math. Anal. Appl. 36 (1971), 693 – 696. · Zbl 0224.30006 · doi:10.1016/0022-247X(71)90049-7 [125] Krzysztof Ciesielski and Tomasz Natkaniec, Algebraic properties of the class of Sierpiński-Zygmund functions, Topology Appl. 79 (1997), no. 1, 75 – 99. · Zbl 0890.26002 · doi: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. · Zbl 0924.26003 [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. · Zbl 1066.03047 [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). · Zbl 1290.26015 [129] Jose A. Conejero, V. Müller, and A. Peris, Hypercyclic behaviour of operators in a hypercyclic \?$$_{0}$$-semigroup, J. Funct. Anal. 244 (2007), no. 1, 342 – 348. · Zbl 1123.47010 · doi:10.1016/j.jfa.2006.12.008 [130] Manuel de la Rosa and Charles Read, A hypercyclic operator whose direct sum \?\oplus \? is not hypercyclic, J. Operator Theory 61 (2009), no. 2, 369 – 380. · Zbl 1193.47014 [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. · Zbl 1163.91007 · doi: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. · Zbl 1154.30004 · doi: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. · Zbl 0542.46007 [134] Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. · Zbl 1034.46504 [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. · Zbl 0565.26003 · doi: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. · Zbl 0036.36303 [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). · Zbl 1297.46022 [138] P.H. Enflo, V.I. Gurariy, and J.B. Seoane-Sepúlveda, On Montgomery’s conjecture and the distribution of Dirichlet sums, Preprint (2012). · Zbl 1385.11058 [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. · Zbl 0715.46007 · doi:10.1093/qmath/40.4.409 [140] Maite Fernández-Unzueta, Zeroes of polynomials on \?_{\infty }, J. Math. Anal. Appl. 324 (2006), no. 2, 1115 – 1124. · Zbl 1114.46035 · doi:10.1016/j.jmaa.2006.01.021 [141] Jesús Ferrer, Zeroes of real polynomials on \?(\?) spaces, J. Math. Anal. Appl. 336 (2007), no. 2, 788 – 796. · Zbl 1161.46024 · doi: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. · Zbl 1145.47045 [143] Jesús Ferrer, A note on zeroes of real polynomials in \?(\?) spaces, Proc. Amer. Math. Soc. 137 (2009), no. 2, 573 – 577. · Zbl 1157.47041 [144] V. P. Fonf, V. I. Gurariy, and M. I. Kadets, An infinite dimensional subspace of \?[0,1] consisting of nowhere differentiable functions, C. R. Acad. Bulgare Sci. 52 (1999), no. 11-12, 13 – 16. · Zbl 0945.26010 [145] James Foran, Fundamentals of real analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 144, Marcel Dekker, Inc., New York, 1991. · Zbl 0744.26004 [146] Лекции по теории аппроксимации в комплексной области, ”Мир”, Мосцощ, 1986 (Руссиан). Транслатед фром тхе Герман бы Л. М. Карташов; Транслатион едитед анд щитх а префаце бы В. И. Белый анд П. М. Тамразов. Диетер Гаиер, Лецтурес он цомплеш аппрошиматион, Бирхäусер Бостон, Инц., Бостон, МА, 1987. Транслатед фром тхе Герман бы Ренате МцЛаугхлин. [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. · Zbl 1054.47008 [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. · Zbl 1221.15006 [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. · Zbl 1207.26006 [150] 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. · Zbl 1202.26006 · doi: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. · Zbl 1207.00028 · doi:10.4169/amer.math.monthly.118.02.167 [152] 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. · Zbl 1262.03100 · doi: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. · Zbl 1158.46008 · doi: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. · Zbl 1210.46019 · doi: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. · Zbl 1201.46027 [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. · Zbl 1115.26003 · doi: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. · Zbl 1218.46014 · doi: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. · Zbl 1213.46013 · doi:10.4064/sm201-1-3 [159] F. J. García-Pacheco, F. Rambla-Barreno, and J. B. Seoane-Sepúlveda, \?-linear functions, functions with dense graph, and everywhere surjectivity, Math. Scand. 102 (2008), no. 1, 156 – 160. · Zbl 1159.46021 · doi:10.7146/math.scand.a-15057 [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. · Zbl 1126.28002 · doi: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.-London-Amsterdam, 1964. · Zbl 1085.26002 [162] Bernard R. Gelbaum and John M. H. Olmsted, Theorems and counterexamples in mathematics, Problem Books in Mathematics, Springer-Verlag, New York, 1990. · Zbl 0702.00005 [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. · Zbl 1085.26002 [164] S. Głab, P. Kaufmann, and L. Pellegrini, Large structures made of nowhere $$L^p$$ functions · Zbl 1314.46034 [165] Szymon Głb, 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. · Zbl 1277.26008 [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. · Zbl 0732.47016 · doi: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. · Zbl 1250.47006 [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. · Zbl 1028.47007 · doi:10.1112/S0024611500012454 [170] Karl Grandjot, Über Grenzwerte ganzer transzendenter Funktionen, Math. Ann. 91 (1924), no. 3-4, 316 – 320 (German). · JFM 50.0213.02 · doi:10.1007/BF01556086 [171] Sophie Grivaux, Construction of operators with prescribed behaviour, Arch. Math. (Basel) 81 (2003), no. 3, 291 – 299. · Zbl 1056.47007 · doi: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. [173] Karl-Goswin Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345 – 381. · Zbl 0933.47003 [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). · Zbl 1076.47005 [175] Karl-G. Grosse-Erdmann and Raymond Mortini, Universal functions for composition operators with non-automorphic symbol, J. Anal. Math. 107 (2009), 355 – 376. · Zbl 1435.30159 · doi:10.1007/s11854-009-0013-4 [176] Karl-G. Grosse-Erdmann and Alfredo Peris Manguillot, Linear chaos, Universitext, Springer, London, 2011. · Zbl 1246.47004 [177] Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, AMS Chelsea Publishing, Providence, RI, 2009. Reprint of the 1965 original. · Zbl 1204.01045 [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. · Zbl 1094.46003 [181] Vladimir I. Gurariy and Lucas Quarta, On lineability of sets of continuous functions, J. Math. Anal. Appl. 294 (2004), no. 1, 62 – 72. · Zbl 1053.46014 · doi: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. · Zbl 0299.54003 · doi:10.1112/jlms/s2-10.1.25 [183] Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. · Zbl 0932.14001 [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. · Zbl 0956.26008 [185] Domingo A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99 (1991), no. 1, 179 – 190. · Zbl 0758.47016 · doi:10.1016/0022-1236(91)90058-D [186] D. Herrero, Hypercyclic operators and chaos, J. Operator Theory 28 (1992), 93-103. · Zbl 0806.47020 [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. · Zbl 0274.47004 · doi:10.1512/iumj.1974.23.23046 [188] Roger A. Horn, Editor’s Endnotes, Amer. Math. Monthly 107 (2000), no. 10, 968 – 969. [189] P. Jiménez-Rodríguez, $$c_0$$ is isometrically isomorphic to a subspace of Cantor-Lebesgue functions, J. Math. Anal. Appl. (to appear). · Zbl 1319.46022 [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. · Zbl 1267.46037 · doi: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). · Zbl 1292.26013 [192] F. B. Jones, Connected and disconnected plane sets and the functional equation \?(\?)+\?(\?)=\?(\?+\?), Bull. Amer. Math. Soc. 48 (1942), 115 – 120. · Zbl 0063.03063 [193] Francis Edmund Jordan, Cardinal numbers connected with adding Darboux-like functions, ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.) – West Virginia University. [194] J.P. Kahane, Baire’s category theorem and trigonometric series, J. Analyse Math. 80 (2000), no. 1, 143-182. · Zbl 0961.42001 [195] Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. · Zbl 1055.43001 [196] Y. Katznelson and Karl Stromberg, Everywhere differentiable, nowhere monotone, functions, Amer. Math. Monthly 81 (1974), 349 – 354. · Zbl 0279.26007 · doi:10.2307/2318996 [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. · Zbl 0528.32001 [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. · Zbl 1097.26006 [199] St. Kierst and D. Szpilrajn, Sur certaines singularités des fonctions analytiques uniformes, Fundamenta Math. 21 (1933), 276-294. · Zbl 0008.07401 [200] Sung S. Kim and Kil H. Kwon, Smooth (\?^{\infty }) but nowhere analytic functions, Amer. Math. Monthly 107 (2000), no. 3, 264 – 266. · Zbl 1031.26023 · doi:10.2307/2589322 [201] Derek Kitson and Richard M. Timoney, Operator ranges and spaceability, J. Math. Anal. Appl. 378 (2011), no. 2, 680 – 686. · Zbl 1219.46005 · doi: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. · Zbl 0907.54036 [203] T.W. Körner, Fourier analysis, Cambridge University Press, Cambridge, 1988. · Zbl 0649.42001 [204] S. Koumandos, V. Nestoridis, Y.-S. Smyrlis, and V. Stefanopoulos, Universal series in \?_{\?>1}\ell ^{\?}, Bull. Lond. Math. Soc. 42 (2010), no. 1, 119 – 129. · Zbl 1195.30055 · doi:10.1112/blms/bdp102 [205] 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. · Zbl 0443.03021 [206] H. Lebesgue, Leçons sur l’intégration et la recherche des fonctions primitives, Gauthier-Willars, 1904. · JFM 54.0257.01 [207] Shamim I. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), no. 2, 384 – 390. , https://doi.org/10.1006/jfan.1996.3093 Fernando León-Saavedra and Alfonso Montes-Rodríguez, Linear structure of hypercyclic vectors, J. Funct. Anal. 148 (1997), no. 2, 524 – 545. · Zbl 0999.47009 · doi: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. · Zbl 0961.47003 [209] Fernando León-Saavedra and Vladimír Müller, Hypercyclic sequences of operators, Studia Math. 175 (2006), no. 1, 1 – 18. · Zbl 1106.47011 · doi:10.4064/sm175-1-1 [210] M. Lerch, Ueber die Nichtdifferentiirbarkeit bewisser Funktionen, J. Reine Angew. Math. 103 (1888), 126-138. · JFM 20.0380.01 [211] B. Levine and D. Milman, On linear sets in space \? 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). [212] Joram Lindenstrauss, On subspaces of Banach spaces without quasicomplements, Israel J. Math. 6 (1968), 36 – 38. · Zbl 0157.43702 · doi:10.1007/BF02771603 [213] Jerónimo López-Salazar Codes, Vector spaces of entire functions of unbounded type, Proc. Amer. Math. Soc. 139 (2011), no. 4, 1347 – 1360. · Zbl 1218.46026 [214] Jerónimo López-Salazar, Lineability of the set of holomorphic mappings with dense range, Studia Math. 210 (2012), no. 2, 177 – 188. · Zbl 1273.46031 · doi: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. · Zbl 1133.46025 · doi: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). · Zbl 0231.30005 [217] Wolfgang Luh, Holomorphic monsters, J. Approx. Theory 53 (1988), no. 2, 128 – 144. · Zbl 0669.30020 · doi: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. · Zbl 0049.05603 [219] F. Martínez-Giménez, P. Oprocha, and A. Peris, Distributional chaos for operators with full scrambled sets, Math. Z. (to appear). · Zbl 1279.47017 [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). · Zbl 1256.30053 · doi:10.4064/sm207-2-5 [221] Q. Menet, Hypercyclic subspaces on Fréchet spaces without continuous norm · Zbl 1301.47009 [222] Q. Menet, Hypercyclic subspaces and weighted shifts · Zbl 1322.47016 [223] Alfonso Montes-Rodríguez, Banach spaces of hypercyclic vectors, Michigan Math. J. 43 (1996), no. 3, 419 – 436. · Zbl 0907.47023 · doi:10.1307/mmj/1029005536 [224] Alfonso Montes-Rodríguez, A Birkhoff theorem for Riemann surfaces, Rocky Mountain J. Math. 28 (1998), no. 2, 663 – 693. · Zbl 0928.30029 · doi: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. · Zbl 1008.47010 · doi: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. · Zbl 1054.47009 · doi: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). · Zbl 1204.42008 · doi: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. · Zbl 1156.30030 · doi: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. · Zbl 0586.46040 [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. · Zbl 0586.46040 [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. · Zbl 1147.46021 · doi:10.1016/j.laa.2008.01.008 [232] T. Natkaniec, Almost continuity, Real Anal. Exchange 17 (1991/92), no. 2, 462 – 520. · Zbl 0760.54007 [233] Tomasz Natkaniec, New cardinal invariants in real analysis, Bull. Polish Acad. Sci. Math. 44 (1996), no. 2, 251 – 256. · Zbl 0876.26004 [234] Vassili Nestoridis, Universal Taylor series, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 5, 1293 – 1306 (English, with English and French summaries). · Zbl 0865.30001 [235] 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. · Zbl 0435.28011 [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. · Zbl 1205.47012 · doi: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. · Zbl 1101.47006 · doi: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. · Zbl 0919.46036 · doi:10.1006/jmaa.1997.5826 [239] Krzysztof Płotka, Sum of Sierpiński-Zygmund and Darboux like functions, Topology Appl. 122 (2002), no. 3, 547 – 564. · Zbl 1005.26001 · doi: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. · Zbl 0527.26003 · doi:10.2307/2975762 [241] Alfred Pringsheim, Ueber die Multiplication bedingt convergenter Reihen, Math. Ann. 21 (1883), no. 3, 327 – 378 (German). · JFM 15.0177.01 · doi: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. · Zbl 1135.47014 · doi: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. · Zbl 0782.47002 · doi:10.1007/BF02765019 [244] David A. Redett, Strongly annular functions in Bergman space, Comput. Methods Funct. Theory 7 (2007), no. 2, 429 – 432. · Zbl 1343.30043 · doi:10.1007/BF03321655 [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. · Zbl 0844.46007 [246] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22. · Zbl 0174.44203 [247] Haskell P. Rosenthal, On quasi-complemented subspaces of Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 361 – 364. · Zbl 0157.43701 [248] Walter Rudin, Holomorphic maps of discs into \?-spaces, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), Springer, Berlin, 1977, pp. 104 – 108. Lecture Notes in Math., Vol. 599. [249] Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. · Zbl 0925.00005 [250] Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. · Zbl 0867.46001 [251] Héctor N. Salas, Supercyclicity and weighted shifts, Studia Math. 135 (1999), no. 1, 55 – 74. · Zbl 0940.47005 [252] Helmut Salzmann and Karl Zeller, Singularitäten unendlich oft differenzierbarer Funktionen, Math. Z. 62 (1955), 354 – 367 (German). · Zbl 0064.29903 · doi:10.1007/BF01180644 [253] Juan B. Seoane, Chaos and lineability of pathological phenomena in analysis, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.) – Kent State University. [254] Juan B. Seoane-Sepúlveda, Explicit constructions of dense common hypercyclic subspaces, Publ. Res. Inst. Math. Sci. 43 (2007), no. 2, 373 – 384. · Zbl 1145.47008 [255] Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. · Zbl 0791.30033 [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. · Zbl 0812.58062 [257] Juichi Shinoda, Some consequences of Martin’s axiom and the negation of the continuum hypothesis, Nagoya Math. J. 49 (1973), 117 – 125. · Zbl 0266.04004 [258] Stanislav Shkarin, On the set of hypercyclic vectors for the differentiation operator, Israel J. Math. 180 (2010), 271 – 283. · Zbl 1218.47017 · doi: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. · Zbl 1263.47010 [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. · Zbl 0551.32013 · doi: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. · JFM 49.0179.02 [262] J. Stallings, Fixed point theorems for connectivity maps, Fund. Math. 47 (1959), 249 – 263. · Zbl 0114.39102 [263] Lynn Arthur Steen and J. Arthur Seebach Jr., Counterexamples in topology, 2nd ed., Springer-Verlag, New York-Heidelberg, 1978. · Zbl 0386.54001 [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. · Zbl 0658.03028 · doi: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. · Zbl 1145.32002 [267] M. Valdivia, Spaces of holomorphic functions in regular domains, J. Math. Anal. Appl. 350 (2009), no. 2, 651-662. · Zbl 1163.30013 [268] Daniel J. Velleman, Characterizing continuity, Amer. Math. Monthly 104 (1997), no. 4, 318 – 322. · Zbl 0871.26002 · doi:10.2307/2974580 [269] Clifford E. Weil, On properties of derivatives, Trans. Amer. Math. Soc. 114 (1965), 363 – 376. · Zbl 0163.29604 [270] Jochen Wengenroth, Hypercyclic operators on non-locally convex spaces, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1759 – 1761. · Zbl 1039.47009 [271] D. J. White, Functions preserving compactness and connectedness are continuous, J. London Math. Soc. 43 (1968), 714 – 716. · Zbl 0159.24502 · doi:10.1112/jlms/s1-43.1.714 [272] A. Wilansky, Semi-Fredholm maps in FK spaces, Math. Z. 144 (1975), 9-12. · Zbl 0294.46011 [273] Gary L. Wise and Eric B. Hall, Counterexamples in probability and real analysis, The Clarendon Press, Oxford University Press, New York, 1993. · Zbl 0827.26001 [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 orders.”, Fund. Math. 36 (1949), 319 – 320 (French). · Zbl 0038.04201 [275] Lawrence Zalcman, Uniqueness and nonuniqueness for the Radon transform, Bull. London Math. Soc. 14 (1982), no. 3, 241 – 245. · Zbl 0464.28006 · doi:10.1112/blms/14.3.241
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