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Algebraic Foundations of Tensor Spaces

  • Wolfgang Hackbusch
Chapter
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 42)

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  • Wolfgang Hackbusch
    • 1
  1. 1.Max-Planck-Institute for Mathematics in the SciencesLeipzigGermany

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