Banach Tensor Spaces

Abstract

The discussion of topological tensor spaces has been started by Schatten [254] and Grothendieck [129, 130]. In Section 4.2 we discuss the question how the norms of V and W are related to the norm of V W: In particular, we introduce the pro- and injective norms. From the viewpoint of functional analysis, tensor spaces of order two are of special interest, since they are related to certain operator spaces (cf. 4.2.9). However, for our applications we are more interested in tensor spaces of order 3. These spaces are considered in Section 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 Section 4.1. Sections 4.4 and 4.5 discuss the case of Hilbert spaces and Hilbert tensor spaces. This is important, since many applications and many of the numerical methods require the Hilbert structure. The reason is that, unfortunately, the solution of approximation problems with respect to general Banach norms is much more involved than those with respect to a scalar product. In Section 4.6 we recall the tensor operations. The final Section 4.7 is devoted to symmetric and antisymmetric tensor spaces.