We consider the cardinal invariant cgx of the minimal number of weakly compact subsets which generate a banach space x. X y is known to be weakly compact precisely when t is continuous from the wright topology to the norm. We are able to answer this question in the following cases. Weakly compact operators between banach spaces mathoverflow. Furhter, a is compact if and only if any net v a has a cluster point in a. Weakly compactly generated locally convex spaces cambridge core. We also show that a weakly compact set embeds in a superreflexive space iff it embeds in hilbert space. Moreover for bounded sets h of a banach space e we consider the worst.
Junnila1 department of mathematics, university of helsinki, yliopistonkatu 5, 00100 helsinki 10, finland communicated by d. Relative weak compactness of solid hulls in banach lattices. 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. It is known that in kbspaces the solid hull of every relatively weakly compact set is again relatively weakly compact see ab2, theorem. Compact sets in banach spaces in a euclidean space, a set is compact if and only if it is closed and bounded. The main reason to prove theorems in this setting is the generality. Compact operators on banach spaces jordan bell jordan. 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. The number of weakly compact sets which generate a banach.
New techniques are introduced to extract subsequences satisfying certain upper estimates in suitable banach spaces. On mappings between banach spaces preserving acompact sets. Weakly compact sets and smooth norms in banach spaces. We study the notion of super weakly compact subsets of a banach space, which can be described as a local version of superreflexivity. One that involves pointwise lower semicontinuous norms and. Embeddings of weakly compact sets and vpaired banach spaces h.
Pdf weakly compact sets and smooth norms in banach spaces. Weakly compact sets the problem of understanding weak compactness in banach spaces is related to many problems in analysis and probability. 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. Pdf on some new characterizations of weakly compact sets. Definition of weakly continuous map from one banach space to. That is, given separable banach spaces x and y and a set of weakly compact operators a. 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. The space c0,1 of continuous functions with the sup norm. This reduces easily to the case in which c is bounded, and will be answered in the affirmative theorem 4.
Making a comparison between a weakly compact set and a superweakly compact set, a further question is arising as follows. Moreover, the closed unit ball in a normed space x is compact in the weak topology if and only if x is reflexive. Then the sets a and b can be strictly separated by a closed hyperplane. Our first result is that the closed convex hull of a super weakly compact set is super weakly compact. Factorization of weakly compact operators between banach. Moreover, an upper bound for the class of sets which are contained in a banach disc is presented. Weakly compact sets in banach spaces are plainly different from general compact hausdorff spaces because they are sequentially compact, and each subset of a. Y, we want to know when does there exist a re exive banach space zsuch that t factors through zfor all t2a.
Using these methods, alternative proofs of results of rakov math. The set a is said to be compactif any cover of a by open sets admits a. Theorem 2 hahnbanach, second geometric version let e be a normed vector space, and let a,b 6. 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. 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. Let xand y be separable banach spaces and let abe a set of weakly. On super weak compactness of subsets and its equivalences. 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. We also show that every eberlein compact space can be embedded as a free generating set in some c 0. X, is a stronger locally convex topology, which may be analogously characterized by taking re. 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. 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. 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.
Embedding of weakly compact sets a,d paired banach spaces. Here is a description of the content of the book, chapter by chapter. This allows us to extend to the non convex setting the main properties of these sets. A note on weakly compact subsets in the projective tensor.
Abstract two smoothness characterizations of weakly compact sets in banach spaces are given. Weakly pcompact, pbanachsaks, and superreflexive banach. 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. M is a weakly compact set in a banach space x and p. We show several characterizations of weakly compact sets in banach spaces.
Joseph diestel the classical theorem of dunford and pettis, identi. 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. Convexcompact sets and banach discs springer for research. An important fact about the weak topology is the banachalaoglu theorem. Weakly compact operators and the strong topology for a banach. Given a bounded closed convex set c of a banach space x, the following. In the second section we show that in a separable reflexive space, every weakly compact set is the intersection of finite unions of cells. Haydon 4 on banach spaces on which every bounded linear operator is the sum of scalar multiple of the identity and a compact operator. 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. Embeddings of weakly compact sets and paired banach spaces. The preimage of weakly open sets being weakly open would also be sufficient, for example. One that involves pointwise lower semicontinuous norms and one that involves projectional resolutions of identity. Given a bounded closed convex set c of a banach space x, the following statements are equivalent. Work by kalton, emmanuele, bator and lewis on the space of compact and weakly compact operators motivates much of this paper.
The classical theorem of dunford and pettis, identifies the. This fails in all infinitedimensional banach spaces and in particular in hilbert spaces where the closed unit ball is not compact. Section 3 is devoted mainly to show some ways to produce relatively pcompact sets in banach spaces not belonging to c p. Thus, polynomials do not preserves weakly compact sets.
Locally convex spaces which are generated by a weakly compact subset or a sequence of weakly compact subsets and their subspaces are studied. Grothendieck 7, who showed the following proposition 2. Pdf we show several characterizations of weakly compact sets in banach spaces. Weakly compact polynomials preserves weakly compact sets. Since xhas a metric and hence has an associated topology, all the standard topological notions openclosed sets, convergence, etc. In the projective tensor product of two banach spaces, the characterization of relatively norm compactness is due to a. In this note, we first give a characterization of super weakly compact convex sets of a banach space x. Pdf on some new characterizations of weakly compact sets in. A banach space x will be regarded as a subspace of its bidual x under the canonical embedding i x. Were going to use the hahnbanach theorem to get some ideas about what sets are open in the weak. A normcontinuous linear map between two banach spaces x and y is also weakly continuous, i. Problem 3 whether every superweakly compact set can be embedded into a superre.
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. In the last few years use of the banach space weak topology has proven. It begins with a section on metric spaces and compact sets. The set a is said to be compact if any cover of a by open sets. Pdf two smoothness characterizations of weakly compact sets in banach spaces are given.
Various characterizations and the permanence properties of these spaces are obtained. 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. 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. Certain results valid for weakly compactly generated banach spaces are extended. Lindenstrauss, weakly compact sets, their topological properties and the banach spaces they generate, annals of math. 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 and smooth norms in banach spaces marian fabian, vicente montesinos and vaclav zizler. Chapter 1 discusses some basic concepts that play a central role in the subject.
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