By Pilar Cembranos
"When do the Lebesgue-Bochner functionality areas comprise a replica or a complemented replica of any of the classical series spaces?" This challenge and the analogous one for vectorvalued non-stop functionality areas have attracted an excessive amount of study task within the final twenty-five years. the purpose of this monograph is to offer a close exposition of the solutions to those questions, supplying a unified and self-contained therapy. It offers quite a few effects, tools and strategies, that are invaluable for any researcher in Banach areas and, often, in practical research. This ebook is written at a graduate pupil point, assuming the fundamentals in Banach house thought.
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Additional info for Banach Spaces of Vector-Valued Functions
Moreover, for 1 < p < ~ it is immediate t h a t the m a p Lp(Pl,X) ~ L;(tto,X) f > hi where oo h = 27p1(An)TXa~ is an isometric isomorphism onto~ too. As an immediate consequence of the preceding results we have: T h e o r e m 1 . 6 . 1 . Let (~2, Z , # ) be an arbitrary positive measure space, let 1 ~ p < oo, let F be a separable Banach space, and assume that Lp(#, X ) contains a copy (respectively, a complemented copy) of F. Then there exists a separable, finite and positive measure space ( Y2o, ~o, #o) such that Lv(#o, X ) contains a copy (respectively, a complemented copy) of F.
This leads us to a new question: if (fn) is a c0-sequence in Lp(tt, X), can we assure t h a t the set of all co E (2 for which (fi~(co)) has a co-subsequence is non null ? Bourgain showed t h a t the answer is "Yes" in the case p = 1, and we will see t h a t for 1 < p < co the answer is "Yes", too. Of course, this provides an answer to the p r o b l e m posed at the beginning of the section: this guarantees t h a t if Lp(/*, X ) contains a copy of Co then X has the s a m e property. It m i g h t be surprising t h a t some results in this section are stated for seminormed spaces.
Of course, the #~'s are finite measures. ~p (see ). )fk) has a c0-subsequence (respectively, an ~l-SUbsequence) in the space Lp(#~, X). 1). Probably these results are well known but we have not been able to find them in the literature. We believe that the application of Souslin operation we give could be useful in some other similar situations. The third goal of the section is showing how the differences between Lp(#) spaces depending on the measure # is purely atomic or not, have a natural 40 1.