Moduleability, algebraic structures, and nonlinear properties

García-Pacheco, Francisco Javier; Pérez-Eslava, C.; Seoane‐Sepúlveda, Juan B.

doi:10.1016/j.jmaa.2010.05.016

Cited by 14 publications

(8 citation statements)

References 16 publications

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“…From the fact that the supports of the f i 's are mutually µ-disjoint one infers that these functions are linearly independent. This together with the equality Concerning spaceability, a number of authors have recently devoted much effort to find large closed subspaces within special subsets of L p (for general o specific measures µ such as the Lebesgue measure or the counting measure), in particular within sets of functions which are p-integrable but not q-integrable for some p, q ∈ (0, +∞], see for instance [11], [30], [31], [39], [40], [41], [42], [54], [55], [57] and [58]. Specially, in [40] ( [58], resp.)…”

Section: P Spacesmentioning

confidence: 99%

“…Other interesting properties -such as algebrability, introduced in [8], additivity, introduced in [78,79] (see also [52]), and moduleability [55]-will not be considered here. Note that if X is an infinite dimensional separable Baire topological vector space then c, the cardinality of the continuum, is the maximal dimension allowed to any vector subspace of X.…”

Section: Introductionmentioning

confidence: 99%

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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%

Section: Introductionmentioning

confidence: 99%

Lineability criteria, with applications

Bernal–González

Cabrera

2014

Journal of Functional Analysis

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“…[3-5, 8-10, 12, 19]). Question 1.1 has also appeared in several recent works (see, e.g., [8,12,13,18]) and, for the last decade, there have been several attempts to partially answer it, although nothing conclusive in relation to the original problem has been obtained so far.…”

Section: Introductionmentioning

confidence: 99%

Basic sequences and spaceability in ℓp spaces

Cariello

Seoane‐Sepúlveda

2014

Journal of Functional Analysis

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“…All the implications in the previous diagram are strict. Specifically, examples of sets that are lineable but not spaceable, coneable but not lineable, lineable but not moduleable, moduleable but not algebrable, and algebrable but not strongly algebrable, can be found, respectively, in [178], [2], [157], and [40].…”

Section: Introduction "Strange" Mathematical Objects Throughout Historymentioning

confidence: 99%

“…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%

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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Moduleability, algebraic structures, and nonlinear properties

Cited by 14 publications

References 16 publications

Lineability criteria, with applications

Lineability criteria, with applications

Basic sequences and spaceability in ℓp spaces

Linear subsets of nonlinear sets in topological vector spaces

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