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In this chapter, we study the basic properties of tensor norms. We begin with the dual norm and this leads naturally to the vital concept of accessibility, which can be thought of as an analogue for tensor norms of the approximation property for spaces. We then meet the various injective and projective norms that can be associated with a tensor norm. Next, we turn our attention to the identification of the duals of the Chevet—Saphar tensor norms in terms of the classes of p-integral operators. In the final section, we meet the Hilbertian tensor norm, which plays a central role in the theory. Grothendieck’s Inequality can now be interpreted as the statement that the Hilbertian tensor norm and the largest injective norm are equivalent. Two new classes of operators emerge: the Hilbertian and the 2-dominated operators. We conclude with Grothendieck’s classification of the natural tensor norms.
KeywordsBanach Space Tensor Product Approximation Property Isometric Embedding Finite Dimensional Space
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