We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
70/2010
On Minimal Subspaces in Tensor Representations
Antonio Falcó and Wolfgang Hackbusch
Abstract
In this paper we introduce and develop the notion of minimal subspaces in the framework of algebraic and topological tensor product spaces. This mathematical structure arises in a natural way in the study of tensor representations. We use minimal subspaces to prove the existence of a best approximation, for any element in a Banach tensor space, by means a tensor given in a typical representation format (Tucker, hierarchical or tensor train). We show that this result holds in a tensor Banach space with a norm stronger that the injective norm and in an intersection of finitely many Banach tensor spaces satisfying some additional conditions. Examples by using topological tensor products of standard Sobolev spaces are given.