Minimal Subspaces

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

Abstract

The notion of minimal subspaces is closely connected with the representations of tensors, provided these representations can be characterised by (dimensions of) subspaces. A separate description of the theory of minimal subspaces can be found in Falcó-Hackbusch [57].

The tensor representations discussed in the later Chapters 8, 11, 12 will lead to subsets \({T_r}, {\mathcal{H}_r}, \mathbb{T}_{\rho}\) of a tensor space. The results of this chapter will prove weak closedness of these sets. Another result concerns the question of a best approxima- tion: is the infimum also a minimum? In the positive case, it is guaranteed that the best approximation can be found in the same set.

For tensors \({\rm{v}}\,\,\epsilon\,\,_{a}\bigotimes_{j=1}^{d}\,\,V_j\) we shall define ‘minimal subspaces’ \(U_{j}^{\hbox{min}} (v) \subset V_j\) in Sects. 6.1-6.4. In Sect. 6.5 we consider weakly convergent sequences \({\rm{v}}_n \rightharpoonup {\rm{v}}\) and analyse the connection between \(U_{j}^{\hbox{min}} ({v}_{n})\) and \(U_{j}^{\hbox{min}} (v)\). The main result will be presented in Theorem 6.24. While Sects. 6.1-6.5 discuss minimal subspaces of algebraic tensors \({\rm{v}}\,\,\epsilon\,\,_{a}{\bigotimes_{j=1}^{d}}\,\,V_j\), Sect. 6.6 investigates \(U_{j}^{\hbox{min}} (v)\) for topological tensors \({\rm{v}}\,\,\epsilon\,\,_{\parallel.\parallel}{\bigotimes_{j=1}^{d}}\,\,V_j\). The final Sect. 6.7 is concerned with intersection spaces.

Keywords

Intersection Space Linear Constraint Dual System Elementary Tensor Tensor Space 
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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