Basic sequences and spaceability in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math> spaces

Cariello, Daniel; Seoane‐Sepúlveda, Juan B.

doi:10.1016/j.jfa.2013.12.011

Cited by 47 publications

(31 citation statements)

References 16 publications

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“…Many more pathologies of this kind can be found in the excellent monograph [59]. See also [2,6,20,41,48,59,71,73] for a long list of pathologies enjoyed by continuous and differentiable functions in R R .…”

Section: 22mentioning

confidence: 99%

Differentiability versus continuity: Restriction and extension theorems and monstrous examples

Ciesielski

Seoane‐Sepúlveda

2018

Bull. Amer. Math. Soc.

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The aim of this expository article is to present recent developments in the centuries old discussion on the interrelations between continuous and differentiable real valued functions of one real variable. The truly new results include, among others, the D n -C n interpolation theorem: For every n-times differentiable f : R → R and perfect P ⊂ R there is a C n function g : R → R such that f P and g P agree on an uncountable set and an example of a differentiable function F : R → R (which can be nowhere monotone) and of compact perfect X ⊂ R such that F (x) = 0 for all x ∈ X while F [X] = X; thus, the map f = F X is shrinking at every point while, paradoxically, not globally. However, the novelty is even more prominent in the newly discovered simplified presentations of several older results, including: a new short and elementary construction of everywhere differentiable nowhere monotone h : R → R and the proofs (not involving Lebesgue measure/integration theory) of the theorems of Jarník: Every differentiable map f : P → R, with P ⊂ R perfect, admits differentiable extension F : R → R and of Laczkovich: For every continuous g : R → R there exists a perfect P ⊂ R such that g P is differentiable. The main part of this exposition, concerning continuity and first order differentiation, is presented in an narrative that answers two classical questions: To what extend a continuous function must be differentiable? and How strong is the assumption of differentiability of a continuous function? In addition, we overview the results concerning higher order differentiation. This includes the Whitney extension theorem and the higher order interpolation theorems related to Ulam-Zahorski problem. Finally, we discuss the results concerning smooth functions that are independent of the standard axioms ZFC of set theory. We close with a list of currently open problems related to this subject.2010 Mathematics Subject Classification.

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Section: 22mentioning

confidence: 99%

Differentiability versus continuity: Restriction and extension theorems and monstrous examples

Ciesielski

Seoane‐Sepúlveda

2018

Bull. Amer. Math. Soc.

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“…A thorough description of the most relevant lineability problems and other related topics can be found in the monograph [4] or in the expository paper [5]. The interested reader may also consult the references [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

A hierarchy in the family of real surjective functions

et al. 2017

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This expository paper focuses on the study of extreme surjective functions in R R . We present several different types of extreme surjectivity by providing examples and crucial properties. These examples help us to establish a hierarchy within the different classes of surjectivity we deal with. The classes presented here are: everywhere surjective functions, strongly everywhere surjective functions, Ä-everywhere surjective functions, perfectly everywhere surjective functions and Jones functions. The algebraic structure of the sets of surjective functions we show here is studied using the concept of lineability. In the final sections of this work we also reveal unexpected connections between the different degrees of extreme surjectivity given above and other interesting sets of functions such as the space of additive mappings, the class of mappings with a dense graph, the class of Darboux functions and the class of Sierpiński-Zygmund functions in R R .

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“…After these definitions were introduced by Aron, Gurariy and Seoane-Sepúlveda in [1], many authors showed that many non-linear special sets can often have these properties. See, for instance, the recent works [2][3][4][5][6][7][8], just to cite some. However, before the publication of [1] some authors already found large linear structures enjoying special properties even though they did not explicitly used terms like lineability or spaceability (see [9,10]).…”

Section: Introductionmentioning

confidence: 99%

The history of a general criterium on spaceability

Sánchez

2017

Open Mathematics

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There are just a few general criteria on spaceability. This survey paper is the history of one of the first ones. Let I 1 and I 2 be arbitrary operator ideals and E and F be Banach spaces. The spaceability of the set of operators I 1 .E; F / n I 2 .E; F / is studied. Before stating the criterium, the paper summarizes the main results about lineability and spaceability of differences between particular operator ideals obtained in recent years. They are the seed of the ideas contained in the general criterium.

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Basic sequences and spaceability in ℓp spaces

Cited by 47 publications

References 16 publications

Differentiability versus continuity: Restriction and extension theorems and monstrous examples

Differentiability versus continuity: Restriction and extension theorems and monstrous examples

A hierarchy in the family of real surjective functions

The history of a general criterium on spaceability

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