ABSTRACT

In this chapter, the author discusses the mathematical structures that combine algebraic and topological properties. Examples of such structures are normed vector spaces, normed rings, seminormed vector spaces and topological vector spaces. The chapter extends the conventional and very useful in mathematics and its applications concepts of norm, seminorm, semimetric, pseudometric and metric utilizing hypernumbers. This allows introduction of extended metric and norm structures into a much larger class of spaces, extending the scope of applications and achieving higher precision in measuring distances, areas and volumes. The chapter summarizes and studies conventional concepts of norms, seminorms, metrics and some of their generalizations.