Maximal cluster sets along arbitrary curves

Bernal–González, Luis; Calderón-Moreno, M. C.; Prado-Bassas, J. A.

doi:10.1016/j.jat.2004.06.003

Cited by 16 publications

(11 citation statements)

References 8 publications

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“…The following theorem due to the Bernal, Calderón and the author [12] extends the result of Kierst and Szpilrajn by setting the large "algebraic" (not only "topological") size of the set of functions f ∈ H(D) with maximal cluster set along any admissible curve. The Nersesjan approximation theorem (see [21] and [34]) is an important ingredient in its proof.…”

Section: Theorem 28 Let G Be a Domain In C Such That Its Boundary ∂mentioning

confidence: 75%

Maximal cluster sets on spaces of holomorphic functions

Prado-Bassas¹,

Antonio²

2008

Analysis

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Section: Theorem 28 Let G Be a Domain In C Such That Its Boundary ∂mentioning

confidence: 75%

Maximal cluster sets on spaces of holomorphic functions

Prado-Bassas¹,

Antonio²

2008

Analysis

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“…In particular, if G = D and t 0 ∈ T is fixed, then we obtain the existence of a residual set of functions with maximal radial cluster set at one prescribed point t 0 ∈ *G. By using Baire's theorem, a residual set of functions f ∈ H (D) can be also obtained such that C (f (j ) , t) = C for all j 0 and all t belonging to a prescribed dense countable subset of T, while Tenthoff [18] provides a dense subset M of H (D) such that C (f (j ) , t) = C for all j 0, all f ∈ M and all t ∈ T. This result can also be obtained as a consequence of [7,Theorem 5] if one takes into account that the polynomials are dense in H (D). Finally, in [6], it is shown, as a special instance of [6, Theorem 2.1], that there is a dense linear manifold of functions with maximal radial cluster sets at any point of T. In addition, it is observed in [6, Section 3] that, as a consequence of Collingwood's maximality theorem, there is a residual subset of functions f ∈ H (D) such that C (f, t 0 ) is maximal for all t 0 belonging to some residual subset of T depending on f. In view of these results, the next question arises:…”

Section: Introduction and Notationmentioning

confidence: 92%

Simultaneously maximal radial cluster sets

Bernal–González

Calderón-Moreno

Prado-Bassas

2005

Journal of Approximation Theory

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“…Concerning domains different from C, Bernal, Calderón, and Prado-Bassas [77] gave in 2004 a linear version of an old theorem due to Kierst and Szpilrajn [199] asserting the residuality of a family of functions holomorphic in G having wild behavior near the boundary. Here G is a Jordan domain in C. Specifically, it is shown in [77] the dense-lineability in H(G) of the set of holomorphic functions f in G satisfying that the cluster set C(f, γ, ξ) of f along γ at each ξ ∈ ∂G equals C ∞ for every f ∈ M \ {0}, every ξ ∈ ∂G and every curve γ ⊂ G tending to ∂G whose closure does not contain ∂G. Recall that if A ⊂ G, then C(f, A, ξ) is defined as the set {w ∈ C ∞ : ∃(z n ) ⊂ A such that z n → ∂G and f (z n ) → w}.…”

Section: General Summability Sequence Spacesmentioning

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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Maximal cluster sets along arbitrary curves

Cited by 16 publications

References 8 publications

Maximal cluster sets on spaces of holomorphic functions

Maximal cluster sets on spaces of holomorphic functions

Simultaneously maximal radial cluster sets

Linear subsets of nonlinear sets in topological vector spaces

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