ABSTRACT
This chapter begins with the description of the classical concept of a topological vector space and only then come to the new structure of a semitopological vector space. Some authors additionally demand that the point 0 in a topological vector space is closed. This condition results in the Hausdorff topology in topological vector spaces. The concept of a semitopological vector space is a natural extension of the concept of a topological vector space. This extension provides a possibility to expand the scope of the theory and its applications while preserving many useful results. The chapter shows that addition is continuous and scalar multiplication is continuous in the second coordinate with respect to this topology. The composition of sequentially continuous mappings is a sequentially continuous mapping. Due to this fact, topological spaces and sequentially continuous mappings form a category.
