By Velasco M. V. (Ed), Rodriguez-Palacios A. (Ed)
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Extra resources for Advanced Courses of Mathematical Analysis 2: Proceedings of the 2nd International School Granada, Spain 20 - 24 September 2004
Xnl)-' so that the weak type (1,l) inequality cannot be satisfied. On the other hand, M s f (x) is always pointwise bounded by the composition M I (M2(. . (Mn f ) . . )) where the Mj are the directional operators defined in the previous section. Each one of these operators is bounded on Lp(R") so that M s is also bounded on LP (R") . As a corollary we can say that the differentiation holds for rectangles with sides parallel to the coordinate axes if f E LEc for some p > 1. This differentiation result had been proved in the 1930's by A.
Lotz) to extend a tensor norm a on FIN to a tensor norm on NORM (see Ref. [ 5 , Ch. II]), that was not considered by Grothendieck. And there is a good reason for that. In fact, if E and F have the approximation property, E @s F = E @= F , and it was 20 years after Grothendieck’s Rbume‘that the first space without the approximation property was discovered! Grothendieck considers some operations with the @-norms on FIN, which are extended to NORM by the finite hull procedure. In particular, when M , N E FIN, for every @-norm a , M @ N = (M’ ma N’)’ (algebraically).
Acad. Sci. Paris 231 (1950), 940-941. (4) Quelques rbultats sur les espaces wectoriels topologiques. C . R. Acad. Sci. Paris 233 (1951)’ 839-841. (5) Sur une notion de produit tensoriel topologique d’espaces wectoriels topologiques, et une classe remarquable d’espaces wectoriels like d cette notion. C . R. cad. ci. Paris 233 (1951), 1556-1558. 35 (6) Critkres de compacite' duns les espacies fonctionnels ge'ne'raux. Amer. J. of Math. 74 (1952), 168-186. (7) Sur les applications line'aires faiblement compactes d'espaces du type C(K).
Advanced Courses of Mathematical Analysis 2: Proceedings of the 2nd International School Granada, Spain 20 - 24 September 2004 by Velasco M. V. (Ed), Rodriguez-Palacios A. (Ed)