Topology in nonlinear extensions of hypernumbers (Q2495102)

From the author's summary: Modern theory of dynamical systems is mostly based on nonlinear differential equations and operations. At the same time, the theory of hypernumbers and extrafunctions, a novel approach in functional analysis, has been limited to linear systems. In this paper, nonlinear structures are introduced in the space of hypernumbers by extending the concept of a hypernumber. In such a way, linear algebras of extended hypernumbers are built. A special topology of conical neighborhoods in these algebras is introduced and studied. It is proved that the space of all extended real hypernumbers is Hausdorff. This provides uniqueness for limits what is very important for analysis of dynamical systems. It is also proved that construction of extended real hypernumbers is defined by a definite invariance principle: the space of all extended real hypernumbers is the biggest Hausdorff factorization of the sequential extension of the space of real numbers with the conical neighborhoods. In addition, this topology turns the set of all bounded extended real hypernumbers into a topological algebra. Other topologies in spaces of extended hypernumbers are considered. Some remarks by the reviewer: 1. The paper deals with real hypernumbers and their extensions. Symbols concerning the complex number case proposed in the Introduction are not used in the text (with one exception in Def. 2.16 (page 150) where two of them are used incorrectly, since the situation described there concerns the real -- not complex -- case). 2. The relation ``\(\leq\)'' defined for sequences of real numbers (Def. 2.46) is not a partial order since it is not true that: \(a\leq b\) and \(b\leq a\) implies \(a= b\), as it is stated in Lemma 2.47. Fortunately, the induced relation ``\(\leq\)'' or hypernumbers and extended hypernumbers (Def. 2.48) is a partial order and so the reasoning based on this fact is correct.