Lineability in sequence and function spaces

Araújo, Gustavo; Bernal–González, Luis; Muñoz-Fernández, Gustavo A.; Prado-Bassas, J. A.; Seoane‐Sepúlveda, Juan B.

doi:10.4064/sm8358-10-2016

Cited by 30 publications

(16 citation statements)

References 21 publications

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“…Since F \ {n} is finite, it follows that Φ n −→ 0 uniformly on X. Therefore, there exists n 0 ∈ N such that |Φ n (x)| < 1 2 for all n ∈ N, n ≥ n 0 and all x ∈ X. Hence, by the reverse triangle inequality, we obtain…”

Section: Now Local Compactness Implies the Existence For Eachmentioning

confidence: 88%

Undominated Sequences of Integrable Functions

et al. 2020

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In this paper, we investigate to what extent the conclusion of the Lebesgue dominated convergence theorem holds if the assumption of dominance is dropped. Specifically, we study both topological and algebraic genericity of the family of all null sequences of functions that, being continuous on a locally compact space and integrable with respect to a given Borel measure in it, are not controlled by an integrable function.

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Section: Now Local Compactness Implies the Existence For Eachmentioning

confidence: 88%

Undominated Sequences of Integrable Functions

et al. 2020

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“…Theorem 5. 3 The subset of ∞ \ c 0 defined over Q p whose elements only have finitely many zero coordinates is strongly c-algebrable.…”

Section: P-adic Sequence Spaces and Failure Of The Cesàro And Hahn-banach Theoremsmentioning

confidence: 99%

Classical vs. non-Archimedean analysis: an approach via algebraic genericity

Sánchez

Maghsoudi

Rodríguez-Vidanes

et al. 2022

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

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In this paper, we show new results and improvements of the non-Archimedean counterpart of classical analysis in the theory of lineability. Besides analyzing the algebraic genericity of sets of functions having properties regarding continuity, discontinuity, Lipschitzianity, differentiability and analyticity, we also study the lineability of sets of sequences having properties concerning boundedness and convergence. In particular we show (among several other results) the algebraic genericity of: (i) functions that do not satisfy Liouville’s theorem, (ii) sequences that do not satisfy the classical theorem of Cèsaro, or (iii) functionals that do not satisfy the classical Hahn–Banach theorem.

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“…To end this paper, we shall prove the following algebrability theorem about the convergence of sequences of S N . For recent results about lineability in function sequence spaces, see [3,14,15]. In order to study the existence of algebras, we endow the space of function sequences (R [0,1] ) N with coordenatewise multiplication.…”

Section: Singular Functions and Differentiabilitymentioning

confidence: 99%

Banach spaces and Banach lattices of singular functions

et al. 2021

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The class of real singular functions on the unit interval, that is, those continuous bounded variation functions having null derivative almost everywhere, is studied from the point of view of lineability. In particular, large closed vector subspaces, large linear algebras and large Banach lattices are found to live, except for zero, inside several subclasses of it. These subclasses are related to the size of the zero set, to nowhere monotonicity, or to the existence of noncritical points. Also the family of continuous functions that are constant on full measure sequences of sets is analyzed from this point of view. Moreover, it is studied what happens in this context when one turns from bounded variation topology to uniform convergence topology.

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Lineability in sequence and function spaces

Cited by 30 publications

References 21 publications

Undominated Sequences of Integrable Functions

Undominated Sequences of Integrable Functions

Classical vs. non-Archimedean analysis: an approach via algebraic genericity

Banach spaces and Banach lattices of singular functions

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