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Extra info for Advanced Courses of Mathematical Analysis 2: Proceedings of the 2nd International School Granada, Spain 20 - 24 September 2004
Of course, the norms E and 7r are reasonable (and duals one of the other). In fact, a norm a defined for every pair of normed spaces is reasonable if and only if E 5 a 5 7r. On the other hand, the @-norms should verify a good functorial p r o p erty: the so called metric mapping property. This means that whenever ui E C(Ei,Fi) (i = 1 , 2 ) , then u1 I8uz E L(E1 @JaE z ,F1 @a F z ) ,with norm I11~111IIuzll. Next, Grothendieck gives a method to construct @-norms with good duality properties: First, he considers a @-norm a defined on the class FIN of all the finite-dimensional Banach spaces.
In reality, any of the above properties are equivalent by duality to a weak compactness criteria in the dual of the space enjoying it. And this is the way that Grothendieck proved that C (K)-spaces enjoy both properties. He also proved that both properties were stable under complemented subspaces, finite products and quotients (a trivial consequence of the mentioned fact that W is a surjective operator ideal). LFrom the inclusions (1) and (2) it is clear that DP implies the RDPP. , E is reflexive.
Ref. [ 11, p. 371, [ 5, Chapter 271) Chapter 3 is devoted to the study of tensor norms on Hilbert spaces. The so called hilbertian tensor norm is introduced by the property that the corresponding linear maps factorize through a Hilbert spaces (in Ref. [ 51 is designed as wz;in Ref. [ 71 is noted as 7 2 ) . The relationships with other tensor norms and the different classes of operators that appear, are studied. In particular, canonical factorization results for mappings C -+ H and H -+ L are obtained.