Pointwise lineability in sequence spaces

Pellegrino, Daniel; Raposo, Anselmo

doi:10.1016/j.indag.2020.12.006

Cited by 8 publications

(5 citation statements)

References 17 publications

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“…• In [18] it is proved the pointwise c-lineability of p (X ) \ q< p q (X ) (where X is any Banach space); taking N := c 00 (X ), we get its infinite pointwise c-dense lineability in p (X ). • In [9] it is proved that the set A 0 of sequences of continuous unbounded and integrable functions in [0, +∞) that goes to zero both in L 1 -norm and uniformly in compacta of [0, +∞) is pointwise c-lineable.…”

Section: Theorem 35 the Family N Bcmentioning

confidence: 99%

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Infinite pointwise lineability: general criteria and applications

Calderón-Moreno,

Gerlach-Mena,

Prado-Bassas

2023

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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In this paper we introduce the concept of infinite pointwise dense lineability (spaceability), and provide a criterion to obtain density from mere lineability. As an application, we study the linear and topological structures within the set of infinite differentiable and integrable functions, for any order $$p \ge 1$$ p ≥ 1 , on $$\mathbb R^N$$ R N which are unbounded in a pre-fixed set.

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Section: Theorem 35 the Family N Bcmentioning

confidence: 99%

“…To be more concrete, Pellegrino and Raposo [18] introduced a pointwise type of lineability as follows:…”

Section: Introductionmentioning

confidence: 99%

Infinite pointwise lineability: general criteria and applications

Calderón-Moreno,

Gerlach-Mena,

Prado-Bassas

2023

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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“…For A a commutative algebra and C a non-empty subset of A, we say that C is pointwise strongly c-algebrable if for every f ∈ C there is a subalgebra B of A which is generated by an algebraically independent set G with #G = c and f ∈ B ⊂ C ∪ {0}. It is worth noting that both the terminology and the notion of pointwise strongly algebrability is inspired by previous works concerning a similar notion for lineability/spaceability (see [7,13]).…”

Section: Proof Of Part (Ii) Of Theorem 14mentioning

confidence: 99%

On sets of extreme functions for Fatou's theorem

Alves¹,

Brito²,

Carando³

2023

Preprint

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Bounded holomorphic functions on the disk have radial limits in almost every direction, as follows from Fatou's theorem. Given a zero-measure set E in the torus T, we study the set of functions such that lim r→1 − f (r w) fails to exist for every w ∈ E (such functions were first constructed by Lusin). We show that the set of Lusin-type functions, for a fixed zero-measure set E, contain algebras of algebraic dimension c (except for the zero function). When the set E is countable, we show also in the several-variable case that the Lusin-type functions contain infinite dimensional Banach spaces and, moreover, contain plenty of c-dimensional algebras. We also address the question for functions on infinitely many variables.

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“…It just so happens that, sometimes, the mother vector x unfortunately does not belong to W . Applications of the mother vector technique with happy endings were studied in [21], where A is said to be pointwise α-lineable (pointwise α-spaceable) if, for every x ∈ A, there is a (closed) α-dimensional subspace W of E such that x ∈ W ⊂ A ∪ {0}. For cardinal numbers α and β, the quite general notions of (α, β)-lineability/spaceability were introduced in [10].…”

Section: Introductionmentioning

confidence: 99%

“…For cardinal numbers α and β, the quite general notions of (α, β)-lineability/spaceability were introduced in [10]. Pointwise α-lineability/spaceability is closely related to (1, α)-lineability/spaceability, but in general these notions are not equivalent (see [21,Example 2.2]). However, for sets of non-injective maps, which happens to be the subject of this paper, these notions are equivalent.…”

Section: Introductionmentioning

confidence: 99%