Tensor Spaces and Numerical Tensor Calculus pp 173-196 | Cite as

# Minimal Subspaces

## 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## Preview

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