# Algebraic Foundations of Tensor Spaces

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## 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*.

## Keywords

Vector Space Tensor Product Space Versus Vector Space Versus Tensor Structure
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## Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012