A century of Sierpiński–Zygmund functions

Ciesielski, Krzysztof; Seoane‐Sepúlveda, Juan B.

doi:10.1007/s13398-019-00726-0

Cited by 19 publications

(9 citation statements)

References 86 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For instance, we can name algebrability and strong algebrability defined in [5,8], respectively. We refer the interested reader to [1,2,[4][5][6][10][11][12][13][14][14][15][16][17][19][20][21][22]24,25,[27][28][29]45] for a current state of the art on this topic.…”

Section: Introductionmentioning

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.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

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.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Let us recall some terminology we shall need throughout this work (which can be found in, for instance, [2–6, 8, 9, 16, 23, 36]). Assume that X is a vector space and α is a cardinal number.…”

Section: Introduction Preliminaries and Backgroundmentioning

confidence: 99%

Lineability, algebrability, and sequences of random variables

Sánchez

Seoane‐Sepúlveda

Trutschnig

2022

Mathematische Nachrichten

View full text Add to dashboard Cite

We show that, when omitting one condition in several well‐known convergence results from probability and measure theory (such as the Dominated Convergence Theorem, Fatou's Lemma, or the Strong Law of Large Numbers), we can construct “very large” (in terms of the cardinality of their systems of generators) spaces and algebras of counterexamples. Moreover, we show that on the probability space false(false[0,1false],scriptBfalse(false[0,1false]false),λfalse)$([0,1],\mathcal {B}([0,1]),\lambda )$ the families of sequences of random variables converging in probability but (i) not converging outside a set of measure 0 or (ii) not converging in arithmetic mean are also “very large”.

show abstract

“…Furthermore, in 2005 [1], he proved that the set of differentiable nowhere monotone functions also contains (except for {0}) infinite dimensional linear spaces. After these seminal works, a lot has been done linking many areas of mathematics, such as Set Theory [8,9,11], Real and Complex Analysis [10,18], Linear and Multilinear Algebra [5], Linear Dynamics [15], or Statistics [12]. Let us recall some terminology we shall need throughout this work (which can be found in [3,4,22,24]).…”

Section: Introductionmentioning

confidence: 99%