Journal ArticleDOI
On bases, finite dimensional decompositions and weaker structures in Banach spaces
William B. Johnson,William B. Johnson,Haskell P. Rosenthal,Haskell P. Rosenthal,M. Zippin,M. Zippin +5 more
TLDR
In this article, the connections between bases and weaker structures in Banach spaces and their duals are investigated, and it is shown that every separable π-structures and finite dimensional Schauder decomposition has a basis.Abstract:
This is an investigation of the connections between bases and weaker structures in Banach spaces and their duals. It is proved, e.g., thatX has a basis ifX* does, and that ifX has a basis, thenX* has a basis provided thatX* is separable and satisfies Grothendieck’s approximation property; analogous results are obtained concerning π-structures and finite dimensional Schauder decompositions. The basic results are then applied to show that every separableℒ
p
space has a basis.read more
Citations
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Journal Article
Ultraproducts in Banach space theory.
TL;DR: The first step into Banach space theory was prepared by the development of the local theory of Banach spaces which goes back to the work of J. Lindenstrauss, A. P. Rosenthal and R. C. James as mentioned in this paper.
Journal ArticleDOI
Factoring weakly compact operators
William J. Davis,William J. Davis,Tadek Figiel,Tadek Figiel,William B. Johnson,William B. Johnson,A Pelczynski,A Pelczynski +7 more
TL;DR: The main result of as mentioned in this paper is that every weakly compact operator between Banach spaces factors through a reflexive Banach space, i.e., it is a Banach-Saks property.
Journal ArticleDOI
Metric Cotype
Manor Mendel,Assaf Naor +1 more
TL;DR: This property of metric cotype is used to prove the following dichotomy: A family of metric spaces F is either almost universal (i.e., contains any finite metric space with any distortion > 1), or there exists α > 0, and arbitrarily large (F) is at least Ω((log )α).
References
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Book
Produits Tensoriels Topologiques Et Espaces Nucleaires
TL;DR: In this paper, Bourbaki implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php).
Book ChapterDOI
Normed Linear Spaces
TL;DR: A(D) as discussed by the authors is a function space with norm ∥ ∥ [Definition I, 3, 1] which defines the topology of major interest in the space; a neighborhood basis of a point x is the family of sets {y: ∥ x - y ∥ ≦ e}.
Journal ArticleDOI
Normed Linear Spaces. By M. M. Day. Pp. 139. DM 28. 1958. (Springer, Berlin)
Journal ArticleDOI
Absolutely summing operators in $ℒ_{p}$-spaces and their applications
J. Lindenstrauss,A. Pełczyński +1 more