Products of Vector Spaces

Abstract

In this chapter we consider a process for forming out of a pair consisting of a right vector space R′ and a left vector space S, a group R′ × S called the direct product of the two spaces. The product R′ × S is a commutative group, but in general there is no natural way of regarding this group as a vector space. Our process does lead to a vector space if one of the factors is a two-sided vector space. We define this concept here, and we remark that, if Δ = Φ is a field, then any left or right space can be regarded, in a trivial fashion, as a two-sided space. This leads to the definition of the Kronecker product of two vector spaces over a field. We also discuss the elements of tensor algebra, and we consider the extension of a vector space over a field Φ to a vector space over a field P containing Φ. Finally we consider the concept of a (non-associative) algebra over a field, and we define the direct product of algebras.