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Theory of Real Hypernumbers Properties and Axiomatization

The theory of hypernumbers is a novel approach in functional analysis aimed at the development of such mathematically correct technique, which allows operations with divergent integrals and series. Although, it resembles nonstandard analysis, there are several distinctions between these theories. For example, while nonstandard analysis changes spaces of real and complex numbers by injecting in to them infinitely small numbers and other nonstandard entities, the theory of extrafunctions does not change the inner structure of spaces of real and complex numbers, but adds to them infinitely big numbers as external objects.The goal of this paper is to develop the theory of hypernumbers in a sufficiently general context. In the second section of this work, basic notions and constructions of the theory of hypernumbers are introduced. This provides better conditions for elaboration of a differential calculus in the hypernumber universe. In the third section, algebraic properties of hypernumbers are studied. It is proved (Theorem 3.1) that the set of all H-real hypernumbers is an ordered linear infinite dimensional space over the field of real numbers. The natural topology of the hypernumber universe is investigated in the fourth section. It is proved (Proposition 4.2) that the space of all H-real hypernumbers is Hausdorff. This provides uniqueness for limits what is very important for analysis. In addition to this, it is proved (Theorem 4.2) that construction of hypernumbers is defined by a definite invariance principle: the space of all real hypernumbers is the biggest Hausdorff factorization of the sequential extension of the space of all real numbers. In the fifth section, hypermetric and hypernoms are introduced. A hypermetric with real number values is a conventional metric. Properties of linear hypernormed spaces and hypermetric spaces are studied. It is proved (Proposition 5.1) that the set of all H-real hypernumbers is a hypermetric space. In addition, problems of completeness and fundamental completeness are considered.