Lineability of the set of holomorphic mappings with dense range

López-Salazar, J.

doi:10.4064/sm210-2-5

Cited by 7 publications

(3 citation statements)

References 3 publications

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“…In the setting of complex holomorphic functions, several authors considered linear and algebraic structures in sets of non‐extendible functions (see [2, 8] and the references therein). Other distinguished properties of holomorphic functions were studied, for example, in [3, 9, 11] or [22].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Holomorphic functions with large cluster sets

Alves

Carando

2021

Mathematische Nachrichten

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We study linear and algebraic structures in sets of bounded holomorphic functions on the ball which have large cluster sets at every possible point (i.e., every point on the sphere in several complex variables and every point of the closed unit ball of the bidual in the infinite dimensional case). We show that this set is strongly frakturc‐algebrable for all separable Banach spaces. For specific spaces including ℓ-0.16emp or duals of Lorentz sequence spaces, we have strongly frakturc‐algebrability and spaceability even for the subalgebra of uniformly continous holomorphic functions on the ball.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Holomorphic functions with large cluster sets

Alves

Carando

2021

Mathematische Nachrichten

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“…In the setting of complex holomorphic functions, several authors considered linear and algebraic structures in sets of non-extendible functions (see [2,8] and the references therein). Other distinguished properties of holomorphic functions were studied, for example, in [3] or [19].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Holomorphic functions with large cluster sets

Alves¹,

Carando²

2019

Preprint

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We study linear and algebraic structures in sets of bounded holomorphic functions on the ball which have large cluster sets at every possible point (i.e., every point on the sphere in several complex variables and every point of the closed unit ball of the bidual in the infinite dimensional case). We show that this set is strongly c-algebrable for all separable Banach spaces. For specific spaces including ℓ p or duals of Lorentz sequence spaces, we have strongly c-algebrability and spaceability even for the subalgebra of uniformly continous holomorphic functions on the ball.

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“…In 1976, Glovebnik [166] and independently Rudin [248] proved that the set D := {f ∈ H(D, X) : f (D) is dense in X} is not empty. Very recently, López-Salazar [214] has been able to demonstrate the lineability of D.…”

Section: Is a Jones Function) If Its Graph Intersects Every Closed Sumentioning

confidence: 98%

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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Lineability of the set of holomorphic mappings with dense range

Abstract: Abstract. Let U be an open subset of a separable Banach space. Let T be the collection of all holomorphic mappings / from the open unit disc DcC into U such that /(D) is dense in U. We prove the lineability and density of T in appropriate spaces for different choices of U.

Cited by 7 publications

References 3 publications

Holomorphic functions with large cluster sets

Holomorphic functions with large cluster sets

Holomorphic functions with large cluster sets

Linear subsets of nonlinear sets in topological vector spaces

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