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

Thumbnail Image
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Springer Nature
Google Scholar
Research Projects
Organizational Units
Journal Issue
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.
CRUE-CSIC (Acuerdos Transformativos 2022)
1. Aizpuru, A., Pérez-Eslava, C., Seoane-Sepúlveda, J.B.: Linear structure of sets of divergent sequences and series. Linear Algebra Appl. 418(2–3), 595–598 (2006). 2. Aizpuru, A., Pérez-Eslava, C., García-Pacheco, F.J., Seoane-Sepúlveda, J.B.: Lineability and coneability of discontinuous functions on R. Publ. Math. Debr. 72(1–2), 129–139 (2008) 3. Araújo, G., Bernal-González, L., Muñoz-Fernández, G.A., Prado-Bassas, J.A., Seoane-Sepúlveda, J.B.: Lineability in sequence and function spaces. Stud. Math. 237(2), 119–136 (2017). 4. Aron, R.M., Bernal González, L., Pellegrino, D.M., Seoane Sepúlveda, J.B.: Lineability: The Search for Linearity in Mathematics Monographs and Research Notes in Mathematics, p. xix + 308. CRC Press, Boca Raton (2016) 5. Aron, R.M., Pérez-García, D., Seoane-Sepúlveda, J.B.: Algebrability of the set of non-convergent Fourier series. Stud. Math. 175(1), 83–90 (2006). 6. Aron, R.M., Gurariy, V.I., Seoane-Sepúlveda, J.B.: Lineability and spaceability of sets of functions on R. Proc. Am. Math. Soc. 133(3), 795–803 (2005). 7. Aron, Richard M., Seoane-Sepúlveda, Juan B.: Algebrability of the set of everywhere surjective functions on C. Bull. Belg. Math. Soc. Simon Stevin 14(1), 25–31 (2007) 8. Bartoszewicz, A., Głab, S.: Strong algebrability of sets of sequences and functions. Proc. Am. Math. Soc. 141(3), 827–835 (2013). 9. Bartoszewicz, A., Bienias,M., Głab, S.: Independent Bernstein sets and algebraic constructions. J.Math. Anal. Appl. 393(1), 138–143 (2012). 10. Bastin, F., Conejero, J.A., Esser, C., Seoane-Sepúlveda, J.B.: Algebrability and nowhere Gevrey differentiability. Isr. J. Math. 205(1), 127–143 (2015). 11. Bernal-González, L., Cabana-Méndez, H.J., Muñoz-Fernández, G.A., Seoane-Sepúlveda, J.B.: On the dimension of subspaces of continuous functions attaining their maximum finitely many times. Trans. Am. Math. Soc. 373(5), 3063–3083 (2020). 12. Bernal-González, L., Conejero, J.A., Murillo-Arcila, M., Seoane-Sepúlveda, J.B.: Highly tempering infinite matrices. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 112(2), 341–345 (2018). 13. Bernal-González, L., Fernández-Sánchez, J., Martínez-Gómez, M.E., Seoane-Sepúlveda, J.B.: Banach spaces and Banach lattices of singular functions. Stud. Math. 260(2), 167–193 (2021). 14. Bernal-González, L., Fernández-Sánchez, J., Seoane-Sepúlveda, J.B., Trutschnig, W.: Highly tempering infinite matrices II: from divergence to convergence via Toeplitz–Silverman matrices. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114(4), 202–10 (2020). 15. Bernal-González, L., Muñoz-Fernández, G.A., Rodríguez-Vidanes, D.L., Seoane-Sepúlveda, J.B.: Algebraic genericity within the class of sup-measurable functions. J. Math. Anal. Appl. 483(1), 123–576 (2020). 16. Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Linear subsets of nonlinear sets in topological vector spaces. Bull. Am.Math. Soc. (N.S.) 51(1), 71–130 (2014). 17. Bonilla, A., Muñoz-Fernández, G.A., Prado-Bassas, J.A., Seoane-Sepúlveda, J.B.: Hausdorff and Box dimensions of continuous functions and lineability. Linear Multilinear Algebra 69(4), 593–606 (2021). 18. Cariello, D., Seoane-Sepúlveda, J.B.: Basic sequences and spaceability in �p spaces. J. Funct. Anal. 266(6), 3797–3814 (2014). 19. Ciesielski, K.C., Gámez-Merino, J.L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Lineability, spaceability, and additivity cardinals for Darboux-like functions. Linear Algebra Appl. 440, 307–317 (2014). 20. Ciesielski, K.C., Seoane-Sepúlveda, J.B.: Differentiability versus continuity: restriction and extension theorems and monstrous examples. Bull. Am. Math. Soc. (N.S.) 56(2), 211–260 (2019). 10.1090/bull/1635 21. Ciesielski, K.C., Seoane-Sepúlveda, J.B.: A century of Sierpinski–Zygmund functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(4), 3863–3901 (2019). 22. Enflo, P.H., Gurariy, V.I., Seoane-Sepúlveda, J.B.: Some results and open questions on spaceability in function spaces. Trans. Am. Math. Soc. 366(2), 611–625 (2014). 23. Fernández-Sánchez, J., Martínez-Gómez, M.E., Seoane-Sepúlveda, J.B.: Algebraic genericity and special properties within sequence spaces and series. Rocky Mountain J. Math (accepted) (2019) (preprint) 24. Fernández-Sánchez, J.,Rodríguez-Vidanes, D.L., Seoane-Sepúlveda, J.B.,Trutschnig,W.:Lineability and integrability in the sense of Riemann, Lebesgue, Denjoy, and Khintchine. J. Math. Anal. Appl. 492(1), 124433–16 (2020). 25. Fernández-Sánchez, J., Rodríguez-Vidanes, D.L., Seoane-Sepúlveda, J.B., Trutschnig, W.: Lineability, differentiable functions and special derivatives. Banach J. Math. Anal. 15(1), 18–22 (2021). 26. Fichtenholz, G., Kantorovich, L.: Sur les opérations dans l’espace des functions bornées. Stud. Math. 5, 69–98 (1934) 27. Gámez-Merino, J.L., Seoane-Sepúlveda, J.B.: An undecidable case of lineability in RR. J. Math. Anal. Appl. 401(2), 959–962 (2013). 28. García, D., Grecu, B.C., Maestre, M., Seoane-Sepúlveda, J.B.: Infinite dimensional Banach spaces of functions with nonlinear properties.Math. Nachr. 283(5), 712–720 (2010). 29. García-Pacheco, F.J., Palmberg, N., Seoane-Sepúlveda, J.B.: Lineability and algebrability of pathological phenomena in analysis. J. Math. Anal. Appl. 326(2), 929–939 (2007). 30. Gouvêa, F.Q.: p-Adic Numbers, An Introduction, Universitext, 2, p. vi + 298. Springer, Berlin (1997). 31. Gurariy, V.I., Quarta, L.: On lineability of sets of continuous functions. J. Math. Anal. Appl. 294(1), 62–72 (2004). 32. Gurarii, V.I.: Subspaces and bases in spaces of continuous functions. Dokl. Akad. Nauk SSSR 167, 971–973 (1966). (Russian) 33. Gurarii, V.I.: Linear spaces composed of everywhere nondifferentiable functions. C. R. Acad. Bulgare Sci. 44(5), 13–16 (1991). (Russian) 34. Jech, T.: Set theory, Springer Monographs in Mathematics, The Third Millennium Edition, Revised and Expanded. Springer, Berlin, pp. xiv+769 (2003) 35. Katok, S.: p-Adic Analysis Compared With Real, Student Mathematical Library, 37, American Mathematical Society, Providence; Mathematics Advanced Study Semesters, University Park, p. xiv+152 (2007) 36. Khodabendehlou, J., Maghsoudi, S., Seoane-Sepúlveda, J.B.: Algebraic genericity and summability within the non-Archimedean setting. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115, 21 (2021). 37. Khodabendehlou, J.,Maghsoudi, S., Seoane-Sepúlveda, J.B.: Lineability and algebrability within p-adic function spaces. Bull. Belg. Math. Soc. Simon Stevin 27(5), 711–729 (2020) 38. Khodabandehlou, J., Maghsoudi, S., Seoane-Sepúlveda, J.B.: Lineability, continuity, and antiderivatives in the non-Archimedean setting. Can. Math. Bull. 64(3), 638–650 (2021). 39. Levine, B., Milman, D.: On linear sets in space C consisting of functions of bounded variation, Russian, with English summary. Commun. Inst. Sci. Math.Méc. Univ. Kharkoff [Zapiski Inst. Mat. Mech.] (4) 16, 102–105 (1940) 40. Mahler, K.: An interpolation series for continuous functions of a p-adic variable. J. Reine Angew. Math. 199, 23–34 (1958). 41. Mahler, K.: p-Adic Numbers and their Functions, Cambridge Tracts in Mathematics, p. xi+320. Cambridge University Press, Cambridge (1981) 42. Perez-Garcia, C., Schikhof, W.H.: Locally Convex Spaces Over Non-Archimedean Valued Fields, Cambridge Studies in Advanced Mathematics, 119, p. xiv+472. Cambridge University Press, Cambridge (2010) 43. Robert, A.M.: A Course in p-Adic Analysis, Graduate Texts inMathematics, 198, pp. xvi+437. Springer, New York (2000) 44. Schikhof, W.H.: Ultrametric Calculus, An Introduction to p-Adic Analysis, Cambridge Studies in Advanced Mathematics, 4, p. viii+306. Cambridge University Press, Cambridge (1984) 45. Seoane-Sepúlveda, J.B.: Chaos and lineability of pathological phenomena in analysis. Thesis (Ph.D.), Kent State University, ProQuest LLC, Ann Arbor, MI, p. 139 (2006) 46. van Rooij, A.C.M.: Non-Archimedean Functional Analysis, Monographs and Textbooks in Pure and Applied Mathematics, 51, pp. x+404. Marcel Dekker, Inc., New York (1978)