Algebraic Foundations of Tensor Spaces

Abstract

Since tensor spaces are in particular vector spaces, we start in Sect. 3.1 with vector spaces. Here, we introduce the free vector space (§3.1.2) and the quotient vector space (§3.1.3) which are needed later. Furthermore, the spaces of linear mappings and dual mappings are discussed in §3.1.4. The core of this chapter is Sect. 3.2 containing the definition of the tensor space. Section 3.3 is devoted to linear and multilinear mappings as well as to tensor spaces of linear mappings. Algebra structures are discussed in Sect. 3.4. Finally, symmetric and antisymmetric tensors are defined in Sect. 3.5.