Mukhammadiev A, Tiwari D, Apaaboah G, Giordano P
University of Vienna, Wien, Austria.
University Grenoble Alpes, Saint-Martin-d'Héres, France.
Mon Hefte Math. 2021;196(1):163-190. doi: 10.1007/s00605-021-01590-0. Epub 2021 Jul 3.
It is well-known that the notion of limit in the sharp topology of sequences of Colombeau generalized numbers does not generalize classical results. E.g. the sequence and a sequence converges and only if . 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 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.
众所周知，在科洛姆博广义数序列的锐拓扑中极限的概念并不能推广经典结果。例如，序列 以及序列 收敛当且仅当 。这有几个深刻的后果，例如在级数研究、解析广义函数、或广义函数积分中的西格玛可加性和经典极限定理方面。这些结果的缺失也与 必然不是一个完备有序集这一事实相关，例如所有无穷小量的集合既没有上确界也没有下确界。我们通过引入超自然数、超序列、闭上确界和闭下确界的概念来给出这些问题的一个解决方案。通过这种方式，我们可以推广关于超序列超极限的所有经典定理。本文探讨了可应用于其他非阿基米德情形的思想。
