Applied Mathematicsematics

Advanced Courses of Mathematical Analysis II: Proceedings of by M. V. Velasco, A. Rodriguez-Palacios PDF

By M. V. Velasco, A. Rodriguez-Palacios

ISBN-10: 981256652X

ISBN-13: 9789812566522

ISBN-10: 9812708448

ISBN-13: 9789812708441

This quantity includes a set of articles via prime researchers in mathematical research. It presents the reader with an intensive evaluation of latest instructions and advances in subject matters for present and destiny study within the box.

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Additional resources for Advanced Courses of Mathematical Analysis II: Proceedings of the Second International School, Granada, Spain, 20 - 24 September 2004

Example text

This was the first property introduced by Grothendieck in the paper we are considering. His motivation was a long 1940’s article by N. Dunford and J. , L1 E P ( W ,V P ) ) . Important consequences followed from this fact. Grothendieck axiomatized the property and called it the Dunford-Pettis Property (DPP for short). He gave several equivalent formulations and proved immediately that the property passes from E’ to E . Since the dual of an L1 space (built over a Radon measure) is an L, space, hence isomorphic t o a C ( K ) space, it is enough to prove that this last space enjoys the D P P for recovering the Dunford and Pettis’ result.

Important consequences followed from this fact. Grothendieck axiomatized the property and called it the Dunford-Pettis Property (DPP for short). He gave several equivalent formulations and proved immediately that the property passes from E’ to E . Since the dual of an L1 space (built over a Radon measure) is an L, space, hence isomorphic t o a C ( K ) space, it is enough to prove that this last space enjoys the D P P for recovering the Dunford and Pettis’ result. This is one of the important results contained in Grothendieck’s memoir.

Grothendieck proved that any continuous linear operator from such a C(K)-space into a separable Banach space, is weakly compact. These spaces are now known as Grothendieck spaces. An internal characterization (obtained also by Grothendieck) is that weak* convergent sequences in the dual space are weakly convergent. Obviously, reflexive spaces are Grothendieck (and they are the only separable Grothendieck spaces). Let us add that J. Bourgain proved in 1983 that H” is a Grothendieck space, and that the property can also be localized: it coincides with the P(C,W ;{co}).

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Advanced Courses of Mathematical Analysis II: Proceedings of the Second International School, Granada, Spain, 20 - 24 September 2004 by M. V. Velasco, A. Rodriguez-Palacios


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