ABSTRACT
In this chapter, the author constructs spaces of hypernumbers and studies their properties as the basic tool for the theory of hypernorms, hyperseminorms and semitopological vector spaces. However, each of the approaches to the hypernumber construction has its advantages. When the author builds hypernumbers from rational numbers, the author uses finite constructive elements, while this is not true for the second construction. Indeed, in contrast to rational numbers, real numbers are inherently infinite because they emerge as a result of an infinite process or as a relation in infinite sets. When the first construction is used, it is not necessary to construct separately real numbers – they automatically emerge as a subclass of real hypernumbers. As many scientists assume, there is no infinity in the real world concluding that there is no need in mathematics of infinite, the author explains that infinity is already inherent to the majority of real numbers, while these numbers are very useful to people.
