Sum of Sierpiński–Zygmund and Darboux like functions

Płotka, Krzysztof

doi:10.1016/s0166-8641(01)00184-5

Cited by 18 publications

(17 citation statements)

References 10 publications

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“…(4) In 1997, Balcerzak, Ciesielski, and Natkaniec showed in [4] that, assuming the set-theoretical condition cov(M) = c (which is true under Martin's Axiom or CH), there exists a Darboux function that is in SZ as well. They prove also that there exists a model of ZFC in which there are no such functions (see, also, [8,9,17]…”

Section: Preliminaries and Backgroundmentioning

confidence: 95%

An undecidable case of lineability inRR

Gámez-Merino

Seoane‐Sepúlveda

2013

Journal of Mathematical Analysis and Applications

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Section: Preliminaries and Backgroundmentioning

confidence: 95%

An undecidable case of lineability inRR

Gámez-Merino

Seoane‐Sepúlveda

2013

Journal of Mathematical Analysis and Applications

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“…In 1997, Balcerzak, Ciesielski, and Natkaniec showed in [32] that, assuming the set-theoretical condition cov(M) = c (which is true under Martin's Axiom or CH), there exists a Darboux function that is in SZ as well. They prove also that there exists a model of the ZFC (Zermelo-Fraenkel Axiom of Choice) in which there are no such functions (see also [125,126,239]). Later, Gámez-Merino, Muñoz-Fernández, Sánchez, and Seoane-Sepúlveda (2010) proved in [149, Theorems 5.6 and 5.10] that the set SZ is c + -lineable and also c-algebrable.…”

Section: Maxima and Minima Of Functions In Rmentioning

confidence: 97%

Linear subsets of nonlinear sets in topological vector spaces

Bernal–González

Pellegrino

Seoane‐Sepúlveda

2013

Bull. Amer. Math. Soc.

195

149

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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. 109 5. Some remarks and conclusions. General techniques 113 About the authors 117 Acknowledgments 118 References 118Theorem 1.2 (Gurariy, 1966 [178]). The set of continuous nowhere differentiable functions on [0, 1] is lineable.Let us also recall that, although the set of everywhere differentiable functions in R is, in itself, an infinite dimensional vector space, in 1966 Gurariy obtained the following analogue of Theorem 1.1.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LINEAR SUBSETS OF NONLINEAR SETS IN TOPOLOGICAL VECTOR SPACES 73 Theorem 1.3 (Gurariy, 1966 [178]). The set of everywhere differentiable functions on [0, 1] is not spaceable in C[0, 1]. Also, there exist closed infinite dimensional subspaces of C[0, 1] all of whose members are differentiable on (0, 1).Somehow, what we are seeing is that what one could expect to be an isolated phenomenon can actually even have a nice algebraic structure (in the form of infinite dimensional subspaces). Unfortunately, and as we mentioned above, the Baire category theorem cannot be employed in the search for large subspaces such as the ones mentioned in the previous results. Let us now provide a more formal and complete definition for the above concepts and some other ones. Definition 1.4 (Lineability and spaceability [22,181,253]). Let X be a topological vector space and M a subset of X. Let μ be a cardinal number.(1) M is said to be μ-lineable (μ-spaceable) if M ∪ {0} contains a vector space (resp. a closed vector space) of dimension μ. At times, we shall refer to the set M as simply lineable or spaceable if the existing subspace is infinite dimensional.(2) We also let λ(M ) be the maximum cardinality (if it exists) of such a vector space. 1 (3) When the above linear space can be chosen to be dense in X, we shall say that M is μ-dense-lineable.Moreover, Bernal introduced in [69] the notion of maximal-lineable (and those of maximal-dense-lineable and maximal-spaceable), meaning that, when keeping the above notation, the dimension of the existing linear space equals dim(X). Besides asking for linear spaces, one could also study other structures, such as algebrability and some related ones, which were presented in [25,27,253]. Definition 1.5. Given a Banach algebra A, a subset B ⊂ A, and two cardinal numbers α and β, we say that:

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“…Let us recall that, besides its intrinsic mathematical interest, this class of Sierpiński-Zygmund functions has attracted the interest of many mathematicians, especially in the area of Real Analysis, since its appearance at the beginning of the 20th century (see, e.g. [8,17,18,38]). In [8] it was shown that there exists a Darboux function that is also in SZ(R) and that there exists a model of ZFC in which there are no such functions.…”

Section: Sierpiński-zygmund Functionsmentioning

confidence: 99%

Algebrability, non-linear properties, and special functions

Bartoszewicz

Głąb

Pellegrino

et al. 2013

Proc. Amer. Math. Soc.

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Abstract. We construct uncountably generated algebras inside the following sets of special functions: (i) Sierpiński-Zygmund functions. (ii) Perfectly everywhere surjective functions. (iii) Nowhere continuous Darboux functions. All conclusions obtained in this paper are improvements of some already known results. Preliminaries and backgroundThis paper is a contribution to the very recent trend, in mathematical analysis, of the search for large algebraic structures (linear spaces or algebras) enjoying what one could call "special" properties. As it has become a usual notion nowadays, given a certain property we say that the subset M of a topological vector space X which satisfies it is µ-lineable (respectively, µ-spaceable) if M ∪ {0} contains a vector space (respectively, closed vector space) of dimension µ (finite or infinite). If M contains an infinite-dimensional (closed) vector space, it shall be simply called lineable (respectively spaceable) for short. These notions of lineability and algebrability were coined by V. I. Gurariy in the early 2000's and first introduced in [4,40].In the recent years many examples of vector spaces of functions on R or C enjoying certain special properties have been constructed. Although these latter notions of lineability and spaceability did not appear until recently, the origins of this theory date back to 1966, when a famous example due to V. I. Gurariy ([30, 31]) showed that there exists an infinite dimensional linear space every non-zero element of which is a continuous nowhere differentiable function on C[0, 1]. (This result was later improved, and even Banach spaces of such functions were constructed.) More recently, many authors got interested in this subject and gave a wide range of examples. For instance, in [4] it was shown that the set of everywhere surjective functions in R is 2 c -lineable (where c denotes the cardinality of R) and that the set of differentiable functions on R which are nowhere monotone is lineable in C(R). These behaviors occur, sometimes, in particularly interesting ways. For example, in [32], Hencl showed that any separable Banach space is isometrically isomorphic to a subspace of C[0, 1] whose non-zero elements are nowhere approximately differentiable and nowhere Hölder. We refer the interested reader to [2, 3, 10-13, 15, 16, 22, 25, 27, 37] for a wider range of results in this topic of lineability and spaceability.Of course, one could go further and not just consider linear spaces but, instead, larger or more complex structures. For instance, in [1] the authors showed that there exists and uncountably generated algebra every non-zero element of which is an everywhere surjective function on C, and 2010 Mathematics Subject Classification. Primary: 15A03; Secondary: 28A20, 46J10.

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Sum of Sierpiński–Zygmund and Darboux like functions

Cited by 18 publications

References 10 publications

An undecidable case of lineability inRR

An undecidable case of lineability inRR

Linear subsets of nonlinear sets in topological vector spaces

Algebrability, non-linear properties, and special functions

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