Abstract
It is well-known that the notion of limit in the sharp topology of sequences of Colombeau generalized numbers widetilde{mathbb {R}} does not generalize classical results. E.g. the sequence frac{1}{n}not rightarrow 0 and a sequence (x_{n})_{nin mathbb {N}} converges if and only if x_{n+1}-x_{n}rightarrow 0. This has several deep consequences, e.g. in the study of series, analytic generalized functions, or sigma-additivity and classical limit theorems in integration of generalized functions. The lacking of these results is also connected to the fact that widetilde{mathbb {R}} is necessarily not a complete ordered set, e.g. the set of all the infinitesimals has neither supremum nor infimum. We present a solution of these problems with the introduction of the notions of hypernatural number, hypersequence, close supremum and infimum. In this way, we can generalize all the classical theorems for the hyperlimit of a hypersequence. The paper explores ideas that can be applied to other non-Archimedean settings.
Highlights
A key concept of non-Archimedean analysis is that extending the real field R into a ring containing infinitesimals and infinite numbers could eventually lead to the solution of non trivial problems. This is the case, e.g., of Colombeau theory, where nonlinear generalized functions can be viewed as set-theoretical maps on domains consisting of generalized points of the non-Archimedean ring R
This orientation has become increasingly important in recent years and it has led to the study of preliminary notions of R
The sharp topology on R is the appropriate notion to deal with continuity of this class of generalized functions and for a suitable concept of well-posedness
Summary
A key concept of non-Archimedean analysis is that extending the real field R into a ring containing infinitesimals and infinite numbers could eventually lead to the solution of non trivial problems This is the case, e.g., of Colombeau theory, where nonlinear generalized functions can be viewed as set-theoretical maps on domains consisting of generalized points of the non-Archimedean ring R. This orientation has become increasingly important in recent years and it has led to the study of preliminary notions of R (cf., e.g., [1,2,3,4,11,15,16,17,25]; see below for a self-contained introduction to the ring of Colombeau generalized numbers R). The ideas presented in the present article, which is self-contained, can surely be useful to explore similar ideas in other non-Archimedean settings, such as [5,6,14,18,23]
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