Basic Properties of Ultrafunctions

  • Vieri BenciEmail author
  • Lorenzo Luperi Baglini
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 85)


Ultrafunctions are a particular class of functions defined on a non- Archimedean field \( \mathbb{R}^{*} \supset \mathbb{R} \). They provide generalized solutions to functional equations which do not have any solutions among the real functions or the distributions. In this paper we analyze systematically some basic properties of the spaces of ultrafunctions.


Ultrafunctions delta function distributions non-Archimedean mathematics non-standard analysis. 


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  1. [1]
    Benci V., Ultrafunctions and generalized solutions, Advanced Nonlinear Studies, vol. 13,(2013), pp. 461–486, ISSN: 1536-1365, arXiv:1206.2257.Google Scholar
  2. [2]
    Benci V., Luperi Baglini L., A model problem for ultrafunctions, to appear, arXiv:1212.1370.Google Scholar
  3. [3]
    Benci, V., Di Nasso, M. Alpha-theory: an elementary axiomatic for nonstandard analysis, Expositiones Mathematicae 21 (2003) pp. 355–386.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    Benci V., An algebraic approach to nonstandard analysis, in: Calculus of Variations and Partial differential equations, (G. Buttazzo, et al., eds.), Springer, Berlin, (1999), 285–326.Google Scholar
  5. [5]
    Benci V., Galatolo S., Ghimenti M., An elementary approach to Stochastic Differential Equations using the infinitesimals, in Contemporary Mathematics, 530, Ultrafilters across Mathematics, American Mathematical Society, (2010), 1–22.Google Scholar
  6. [6]
    Benci V., Horsten H.,Wenmackers S., Non-Archimedean probability, Milan J. Math., (2012), arXiv:1106.1524Google Scholar
  7. [7]
    Keisler H.J., Foundations of Infinitesimal Calculus, Prindle, Weber & Schmidt, Boston, (1976). [This book is now freely downloadable at:]

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di PisaPisaItaly
  2. 2.Department of Mathematics College of ScienceKing Saud UniversityRiyadhSaudi Arabia

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