Advanced Courses of Mathematical Analysis 2: Proceedings of by Velasco M. V. (Ed), Rodriguez-Palacios A. (Ed)

By Velasco M. V. (Ed), Rodriguez-Palacios A. (Ed)

This quantity includes a set of articles by way of major researchers in mathematical research. It offers the reader with an in depth review of latest instructions and advances in themes for present and destiny examine within the box.

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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

Example text

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.

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