2012 | OriginalPaper | Buchkapitel
Minimal Subspaces
verfasst von : Wolfgang Hackbusch
Erschienen in: Tensor Spaces and Numerical Tensor Calculus
Verlag: Springer Berlin Heidelberg
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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.