On Minimal Subspaces in Tensor Representations

Antonio Falcó and Wolfgang Hackbusch

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Submission date: 18. Nov. 2010
Pages: 30
published in: Foundations of computational mathematics, 12 (2012) 6, p. 765-803 
DOI number (of the published article): 10.1007/s10208-012-9136-6
Bibtex
MSC-Numbers: 15A69, 46B28, 46A32
Keywords and phrases: numerical tensor calculus, tensor product, tensor space, minimal subspaces, weak closedness
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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.