A Topological Approach to Non-Archimedean Mathematics

Abstract

Non-Archimedean mathematics (in particular, nonstandard analysis) allows to construct some useful models to study certain phenomena arising in PDE's; for example, it allows to construct generalized solutions of differential equations and variational problems that have no classical solution. In this paper we introduce certain notions of non-Archimedean mathematics (in particular, of nonstandard analysis) by means of an elementary topological approach; in particular, we construct non-Archimedean extensions of the reals as appropriate topological completions of R. Our approach is based on the notion of Λ-limit for real functions, and it is called Λ-theory. It can be seen as a topological generalization of the α-theory presented in [6], and as an alternative topological presentation of the ultrapower construction of nonstandard extensions (in the sense of [22]). To motivate the use of Λ-theory for applications we show how to use it to solve a minimization problem of calculus of variations (that does not have classical solutions) by means of a particular family of generalized functions, called ultrafunctions.

Generalized solutions in PDEs and the Burgers' equation

“…In this section we recall the basic definitions and facts regarding non-Archimedean fields, following an approach that has been introduced in [13] (see also [4,6,7,8,9,10,11,12]). In the following, K will denote an ordered field.…”

“…In this section we introduce a particular non-Archimedean field by means of Λ−theory 1 , in particular by means of the notion of Λ−limit (for complete proofs and for further informations the reader is referred to [2], [4], [7] and [13]). To recall the basics of Λ−theory we have to recall the notion of superstructure on a set (see also [22]):…”

“…1 Readers expert in nonstandard analysis will recognize that Λ-theory is equivalent to the superstructure constructions of Keisler (see [22] for a presentation of the original constructions of Keisler, and [13] for a comparison between these two approaches to nonstandard analysis).…”

“…Ultrafunctions are a family of generalized functions defined on the field of hyperreals, which are a well known extension of the reals. They have been introduced in [4], and also studied in [7,8,10,11,12,13]. The non-Archimedean setting in which we will work (which is a reformulation, in a topological language, of the ultrapower approach to NSA of Keisler) is introduced in Section 2.…”

Towards a Morse theory on Banach spaces via ultrafunctions

“…We underline that this material is not original but we cite it in order to make the article (almost) self contained. We refer to [2,3,4,5] for a more detailed treatment.…”