Large structures in certain subsets of Orlicz spaces

Akbarbaglu, Ibrahim; Maghsoudi, S.

doi:10.1016/j.laa.2013.01.038

Cited by 11 publications

(8 citation statements)

References 24 publications

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“…But the case p > 0 is not covered because, to the best of our knowledge, Theorem 3.1 has not been given a proof when the Z n 's are just F-spaces. Nevertheless, by using [30] has been recently used by Akbarbaglu and Maghsoudi [2] to discover spaceability in certain related subsets of Orlicz spaces.…”

Section: P Spacesmentioning

confidence: 99%

Lineability criteria, with applications

Bernal–González

Cabrera

2014

Journal of Functional Analysis

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Lineability is a property enjoyed by some subsets within a vector space X. A subset A of X is called lineable whenever A contains, except for zero, an infinite dimensional vector subspace. If, additionally, X is endowed with richer structures, then the more stringent notions of dense-lineability, maximal dense-lineability and spaceability arise naturally. In this paper, several lineability criteria are provided and applied to specific topological vector spaces, mainly function spaces. Sometimes, such criteria furnish unified proofs of a number of scattered results in the related literature. Families of strict-order integrable functions, hypercyclic vectors, non-extendable holomorphic mappings, Riemann non-Lebesgue integrable functions, sequences not satisfying the Lebesgue dominated convergence theorem, nowhere analytic functions, bounded variation functions, entire functions with fast growth and Peano curves, among others, are analyzed from the point of view of lineability.

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Section: P Spacesmentioning

confidence: 99%

Lineability criteria, with applications

Bernal–González

Cabrera

2014

Journal of Functional Analysis

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“…Before finishing this section, let us point out that (very recently) some results related to the problems treated in this section have been obtained by Barroso, Botelho, Fávaro, and Pellegrino in [36], and by G lab, Kaufmann, and Pellegrini [164,165]. Within the framework of Orlicz spaces, we also refer the reader to the very recent work by Akbarbaglu and Maghsoudi [4]. One of the key points that led Lebesgue to his theory of integration was the existence of two examples, one given by Volterra in 1881 and another by Brodén in 1896 which showed that, at least from the point of view of Riemann integration, the process of obtaining antiderivatives of a function and integration theory were not equivalent.…”

Section: Measurability and Integrationmentioning

confidence: 92%

“…Other types of structures have also been considered, such as cones 3 or modules. 4 The links between the previous concepts are as follows:…”

Section: Introduction "Strange" Mathematical Objects Throughout Historymentioning

confidence: 99%

“…3 A set of functions in R R (or C C ) is said to be coneable if it possesses a positive (or negative) cone containing an infinite linearly independent set; see [2,Definition 1.1]. 4 Let L be a subset of a Banach algebra (or a topological algebra) X. We say that L is moduleable if there exists an infinitely generated subalgebra M of X and an infinitely generated additive subgroup G of X such that G is an (M, K)-bimodule, G is K infinite dimensional, and L∪{0} ⊃ G. For more information on this notion, we refer to [157,Definition 1.2].…”

Section: Introduction "Strange" Mathematical Objects Throughout Historymentioning

confidence: 99%

See 1 more Smart Citation

Linear subsets of nonlinear sets in topological vector spaces

Bernal–González

Pellegrino

Seoane‐Sepúlveda

2013

Bull. Amer. Math. Soc.

197

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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“…We will focus on the outcomes obtained by the authors of this chapter with other coworkers. It should be mentioned here that recently many other authors have considered and got interesting results in the algebrability (see [2], [3], [25]).…”

Section: Introductionmentioning

confidence: 95%