That is, given separable banach spaces x and y and a set of weakly compact operators a. Moreover, the closed unit ball in a normed space x is compact in the weak topology if and only if x is reflexive. Banach function spaces is a very general class of banach spaces including all l p spaces for 1 p 1, orlicz spaces, and orliczlorentz spaces as typical examples. We show several characterizations of weakly compact sets in banach spaces. By means of super weakly compact set theory established in the recent years, in this paper, we first show that a nonempty closed bounded convex set of a banach space has super fixed point property. Here is a description of the content of the book, chapter by chapter.
Embeddings of weakly compact sets and vpaired banach spaces h. M is a weakly compact set in a banach space x and p. Weakly compact sets and smooth norms in banach spaces. Weakly compact sets in banach spaces are plainly different from general compact hausdorff spaces because they are sequentially compact, and each subset of a weakly compact set has a closure that is sequentially determined. Certain results valid for weakly compactly generated banach spaces are extended. Weakly pcompact, pbanachsaks, and superreflexive banach. Compact operators on banach spaces jordan bell jordan. This reduces easily to the case in which c is bounded, and will be answered in the affirmative theorem 4. Convexcompact sets and banach discs springer for research. Pdf two smoothness characterizations of weakly compact sets in banach spaces are given. Weakly compact operators and the strong topology for a banach. Pdf on some new characterizations of weakly compact sets. On mappings between banach spaces preserving acompact sets.
Let xand y be separable banach spaces and let abe a set of weakly. Let lx,y be the banach space of all bounded linear operators between banach spaces x and y, kx,y be the space of all. The number of weakly compact sets which generate a banach. Weakly compact sets in banach spaces are plainly different from general compact hausdorff spaces because they are sequentially compact, and each subset of a. Lx,y between banach spaces is weakly compact if it maps the closed unit ball of x into a weakly relatively compact subset of y. Grothendieck 7, who showed the following proposition 2. Jamesi it has been conjectured that a closed convex subset c of a banach space b is weakly compact if and only if each continuous linear functional on b attains a maximum on c 5. This allows us to extend to the non convex setting the main properties of these sets. Weakly compact sets and smooth norms in banach spaces article pdf available in bulletin of the australian mathematical society 6502 april 2002 with 154 reads how we measure reads. Weakly compact polynomials preserves weakly compact sets. James1 it has been conjectured that a closed convex subset c of a banach space b is weakly compact if and only if each continuous linear functional on b attains a maximum on c 5. Relative weak compactness of solid hulls in banach lattices. Our first result is that the closed convex hull of a super weakly compact set is super weakly compact. One that involves pointwise lower semicontinuous norms and.
The main reason to prove theorems in this setting is the generality. Weakly compact sets and smooth norms in banach spaces marian fabian, vicente montesinos and vaclav zizler. We study the behavior of this index when passing to subspaces, its relation with the lindelof number in the weak topology and other related questions. The definition of a super weakly compact set was introduced in 11 for convex sets it terms of finite representability, in a very similar way as the one for banach spaces, see 5 or 16. Were going to use the hahnbanach theorem to get some ideas about what sets are open in the weak. We consider the cardinal invariant cgx of the minimal number of weakly compact subsets which generate a banach space x. The space c0,1 of continuous functions with the sup norm.
In the projective tensor product of two banach spaces, the characterization of relatively norm compactness is due to a. Thus, polynomials do not preserves weakly compact sets. Then the sets a and b can be strictly separated by a closed hyperplane. On super weak compactness of subsets and its equivalences. Work by kalton, emmanuele, bator and lewis on the space of compact and weakly compact operators motivates much of this paper. Junnila1 department of mathematics, university of helsinki, yliopistonkatu 5, 00100 helsinki 10, finland communicated by d. Factorization of weakly compact operators between banach. We also show that every eberlein compact space can be embedded as a free generating set in some c 0.
Weakly compactly generated locally convex spaces cambridge core. The set a is said to be compactif any cover of a by open sets admits a. Using these methods, alternative proofs of results of rakov math. Since xhas a metric and hence has an associated topology, all the standard topological notions openclosed sets, convergence, etc. It follows from the hahnbanach separation theorem that the weak topology is hausdorff, and that a normclosed convex subset of a banach space is also weakly closed. We are able to answer this question in the following cases. We also show that a weakly compact set embeds in a superreflexive space iff it embeds in hilbert space. In this dissertation we study the structure of spaces of operators, especially the space of all compact operators between two banach spaces x and y. X, is a stronger locally convex topology, which may be analogously characterized by taking re. This fails in all infinitedimensional banach spaces and in particular in hilbert spaces where the closed unit ball is not compact. The set a is said to be compact if any cover of a by open sets. We study the notion of super weakly compact subsets of a banach space, which can be described as a local version of superreflexivity. X y is known to be weakly compact precisely when t is continuous from the wright topology to the norm. Compact sets in banach spaces in a euclidean space, a set is compact if and only if it is closed and bounded.
Lindenstrauss, weakly compact sets, their topological properties and the banach spaces they generate, annals of math. Chapter 1 discusses some basic concepts that play a central role in the subject. Embedding of weakly compact sets a,d paired banach spaces. In the second section we show that in a separable reflexive space, every weakly compact set is the intersection of finite unions of cells. A normcontinuous linear map between two banach spaces x and y is also weakly continuous, i. New techniques are introduced to extract subsequences satisfying certain upper estimates in suitable banach spaces. It has been conjectured that a closed convex subset c of a banach space b is weakly compact if and only if each continuous linear func tional on b attains a. Joseph diestel the classical theorem of dunford and pettis, identi. Locally convex spaces which are generated by a weakly compact subset or a sequence of weakly compact subsets and their subspaces are studied. An important fact about the weak topology is the banachalaoglu theorem.
Moreover, an upper bound for the class of sets which are contained in a banach disc is presented. One that involves pointwise lower semicontinuous norms and one that involves projectional resolutions of identity. Abstract two smoothness characterizations of weakly compact sets in banach spaces are given. Haydon 4 on banach spaces on which every bounded linear operator is the sum of scalar multiple of the identity and a compact operator. In the last few years use of the banach space weak topology has proven. Pdf weakly compact sets and smooth norms in banach spaces. Moreover for bounded sets h of a banach space e we consider the worst. Problem 3 whether every superweakly compact set can be embedded into a superre. Various characterizations and the permanence properties of these spaces are obtained.
A note on weakly compact subsets in the projective tensor. We give an example of a weakly compact set in a banach space, which does not embed topologically as a weakly compact subset of hilbert space. The classical theorem of dunford and pettis, identifies the. Y, we want to know when does there exist a re exive banach space zsuch that t factors through zfor all t2a. Moussa from a banach lattice e into a banach lattice f is called order m weakly compact if for every disjoint sequence xn in be and every order bounded sequence fn of f. Weakly compact sets the problem of understanding weak compactness in banach spaces is related to many problems in analysis and probability. Furhter, a is compact if and only if any net v a has a cluster point in a. These results are obtained as special cases of properties of paired banach spaces, a notion generalizing the relation of a reflexive banach space and its dual. A banach space x will be regarded as a subspace of its bidual x under the canonical embedding i x. The two sections of this note are independent, but they are related by the fact that both use the results of 5 to obtain information on the properties of weakly compact sets in banach spaces.
Given a bounded closed convex set c of a banach space x, the following. Making a comparison between a weakly compact set and a superweakly compact set, a further question is arising as follows. Pdf we show several characterizations of weakly compact sets in banach spaces. Given a bounded closed convex set c of a banach space x, the following statements are equivalent. It begins with a section on metric spaces and compact sets. Theorem 2 hahnbanach, second geometric version let e be a normed vector space, and let a,b 6. Definition of weakly continuous map from one banach space to.
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