Banach Tensor Spaces

Abstract

The discussion of topological tensor spaces has been started by Schatten [167] and Grothendieck [79, 80]. In Sect. 4.2 we discuss the question how the norms of V and W are related to the norm of $V \otimes W.$

From the viewpoint of functional analysis, tensor spaces of order 2 are of particular interest, since they are related to certain operator spaces (cf. §4.2.13). However, for our applications we are more interested in tensor spaces of order ≥ 3. These spaces are considered in Sect. 4.3.

As preparation for the aforementioned sections and later ones, we need more or less well-known results from Banach space theory, which we provide in Sect. 4.1.

Section 4.4 discusses the case of Hilbert spaces. This is important, since many applications are of this kind. Many of the numerical methods require scalar products. The reason is that, unfortunately, the solution of approximation problems with re- spect to general Banach norms is much more involved than those with respect to a scalar product.