A set is crcompact if it is the union of countably many compact sets. The norm closed twosided ainvariant linear subspaces of a. Projections in weakly compactly generated banach spaces and. Recently, deville and the third author 1 have proved that the answer is a rmative when x is hilbert generated, thus improving the previous results obtained in 2, 16. We address the questions if these can be the only projections in a.
A banach space x is called weakly compactly generated wcg if there is a weakly compact set k in x such that x span. Some banach spaces of vectorvalued functions which are weakly compactly generated. Pettis integrability american mathematical society. Weakly compactly generated banach spaces and some of their relatives classi. Weakly compactly generated banach spaces, changs conjecture, few operators.
In particular, if x is either separable or reaexive then it is wcg see for example 7. Projections in dual weakly compactly generated banach spaces to appear studia math. Some equivalent properties of the existence of frechet smooth norms in weakly compactly generated wcg banach spaces are shown. Pdf on equivalent characterizations of weakly compactly. An example of a banach space x with gateaux smooth norm that has no equivalent msmooth norm for any linearly dense set m in x is a non weakly compactly generated subspace of a weakly compactly generated space of density. Q x be a weakly measurable function and assume there is a sub. If a wcg banach space x admits a csmooth function with bounded support, then x admits csmooth partitions of unity. A banach space f is weakly compactly generated if f contains a weakly compact subset whose linear span is dense in f. In mathematics, more specifically in functional analysis, a banach space pronounced is a complete normed vector space. Eudml a function space cx which is weakly lindelof but. The heredity problem for weakly compactly generated banach spaces haskell p.
Introduction a real banach space x is weakly compactly generated wcg if there is a weakly compact subset k c x with x sp k, the norm closed linear hull of k. Although it remains open in general, an affirmative answer is known in many special cases, including 1 reflexive banach spaces, 2 weakly compactly generated wcg spaces, and 3 banach spaces. Much of the work on the geometry of banach spaces was related to wcg spaces. Corollary 7 can also be obtained from the following theorem, essentially due to mercourakis 8. Keywords banach space bounded linear operator closed subspace separable banach space reflexive banach space. Pdf banach spaces with projectional skeletons semantic. Let e be a weakly compactly generated frechet space.
Classical results on weakly compactly generated wcg banach spaces imply the existence of projectional resolutions of identity pri and the existence of many projections on separable subspaces scp. Thus, a banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a cauchy sequence of vectors always converges to a well defined limit that is within the space. These kinds of results respond to a widely treated question in the literature see 2, 3. On equivalent characterizations of weakly compactly generated banach spaces article pdf available in rocky mountain journal of mathematics 244 december 1994 with 18. If a wcg banach space x admits a cksmooth function. Diestel of whether every banach lattice which is generated, as a lattice, by a weakly compact set must be weakly compactly generated i.
Approximation by lipschitz, cp smooth functions on weakly. X has mazurs property if the sequentially weak continuous functionals on x are in x. Also, a conjecture is formulated at the end of this introduction. Talagrand compact iff it is homeomorphic to a weakly compact subset of a banach space resp. Weakly compact sets and smooth norms in banach spaces. Citeseerx document details isaac councill, lee giles, pradeep teregowda. In this note we extend this result to the case when e is a weakly compactly generated. The mcshane integral in weakly compactly generated spaces. Some authors also say that a k kspace is compactly generated, while others reserve that term for a k kspace which is also weak hausdorff, meaning that the image of any t.
Weakly compactly generated banach spaces and the strong. The ideal of weakly compactly generated operators acting on a banach space tomasz kania and tomasz kochanek communicated by kenneth r. We give characterizations of weakly compactly generated spaces, their subspaces, vasak spaces, weakly lindelof determined spaces as well as several other classes of banach spaces related to uniform gateaux smoothness, in terms of the presence of a total subset of the space with some additional properties. Were going to use the hahnbanach theorem to get some ideas about what sets are open in the weak. We recall that a banach space is wcg weakly compactly generated if there exists a weakly compact subset w of e that spans a dense linear space in e one can always. We address the questions if these can be the only projections in a nonseparable wcg space, in the sense. We refer to the last three chapters of 9 for this topic. Smoothness and its equivalents in weakly compactly generated banach spaces to appearj. One thing is that compact generation for weakly hausdorff spaces still takes the simple form that the space is the colimit of its compact subsets. Why the w in cgwh compactly generated weakly hausdorff. X is said to be weakly compactly generated wcg if there exists a weakly compact set k. A characterization of subspaces of weakly compactly generated.
A space x x is a k k space if any hence all of the above conditions hold. If e is a weakly compactly generated banach space then it is proved in 7 that e, with weak topology, is lindelf. Approximation of lipschitz functions bylipschitz, generated. A real banach space x in short a b space x is weakly compactly generated wcg if there is an absolutely convex weakly compact set k c x such that x sp k the closed linear hull of k. Smoothness and its equivalents in weakly compactly generated. Theorem 2 hahnbanach, second geometric version let e be a normed vector space, and let a,b 6. We show that every space with a projectional skeleton has a projectional resolution of.
Get a printable copy pdf file of the complete article 302k, or click on a page image below to browse page by page. A banach space x is called weakly compactly generated wcg if there exists. X is said to be weakly compactly generated in short, wcg if x is the closed linear span of a weakly. The heredity problem for weakly compactly generated banach.
A function space cx which is weakly lindelof but not weakly compactly generated. Ifb is a weakly compactly generated banach space andf. Weakly compactly generated banach spaces and some of. Diestel, geometry of banach spaces, selected topics, lect. Introduction the purpose of this paper is to introduce the free banach lattice generated by a banach space and investigate its properties. Recall that a banach space e is called weakly compactly generated. The free banach lattice over a banach space is introduced and analyzed. To find conditions on a space e such that every linear continuous. We study strongly asplund generated sag and strongly conditionally weakly compactly generated scwcg banach spaces.
The operator idealof weakly compactly generatedoperators and its relations to other operatorideals recall that wcge,f denotes the set of all operators t. Full text full text is available as a scanned copy of the original print version. Inner characterizations of weakly compactly generated. Since the seminal paper by amir and lindenstrauss 1, where they showed the interplay between topological and geometrical properties of the socalled weakly compactly generated banach spaces, a. The free banach lattice generated by a banach space. If a banach space v is a subspace of a weakly compactly generated space and the density character of v is less than b, then v is weakly compactly generated. Recall that a banach space e is called weakly compactly generated if it admits a. Smoothness in weakly compactly generated banach spaces core. Inner characterizations of weakly compactly generated banach.
Weakstar convergence of convex sets cornell university. Projections in weakly compactly generated banach spaces and changs conjecture. Johnson and others published correction to some remarks on weakly compactly generated banach spaces. The banach spacex isweakly compactly generated wcg if there exists a weakly compact subset of x whose linear span is dense in x. But avoid asking for help, clarification, or responding to other answers. Pdf weakly compactly generated banach lattices semantic scholar. We need the following characterization of wcg spaces. Then the sets a and b can be strictly separated by a closed hyperplane. A banach space x is said to admit a separable projectional resolution of the identity spri, if for the. Moreover, an example of a banach lattice which is weakly compactly generated as a lattice but not as a banach space is exhibited, thus answering a question of j. Recently, deville and the third author 1 have proved that the answer is af. Access full article top access to full text full pdf how to cite top. Then e,ge,e is a lindelf space and e is a borel subset of en,gem,e, em being the bidual of e.
Weak sequential compactness and bornological limit derivatives. C x t\colon c\to x is closed when c c is compact hausdorff. A locally convex space is said to be weakly compactly generated if there exists an increasing sequence of ge,ecompact subsets of e whose union is dense in e. Pdf a characterization of subspaces of weakly compactly. Smoothness in weakly compactly generated banach spaces. Zizler shows that x is then isomorphic to a hilbert space.
Preliminaries in the sequel e will be a real banach space which is weakly compactly generated. We address the questions if these can be the only projections in a nonseparable wcg space, in the sense that there is a pri p. The heredity problem for weakly compactly generated banach spaces. A topological space is said to be lindelf if its every open covering has a countable subcovering. Dec 29, 2015 weakly compact set must be weakly compactly generated i. Separable banach spaces are weakly compactly generated and in fact norm compactly generated take a dense sequence xn in the unit sphere and consider f0gfxnn. These spaces have regularity properties not found in a general banach space. Weakly compactly generated banach spaces springerlink. Todor cevi c for explaining that the earlier versions of this paper from march 2000 should be rewritten using the notion of a wcg banach space rather than previous quite technical condition. On countable tightness and the lindelof property in non. Dec 01, 2005 classical results on weakly compactly generated wcg banach spaces imply the existence of projectional resolutions of identity pri and the existence of many projections on separable subspaces scp. Strongly asplund generated and strongly conditionally.
Mazur separable quotient problem asks whether every infinitedimensional banach space has a quotient banach space that is both infinitedimensional and separable. A banach space x is called weakly compactly generated wcg if there exists a weakly compact subset cof xsuch that the linear span of cis dense in x. Thanks for contributing an answer to mathematics stack exchange. A banach space x is said to be weakly compactly generated wcg if there is a weakly compact subset k of x whose linear span is dense in x. Weakly compactly generated banach spaces and some of their.
We call a bounded linear operator acting between banach spaces weakly compactly generated wcg for short if its range is contained in a weakly compactly generated subspace of its target space. Weakly compactly generated banach spaces and the strong law. A space x x is a k kspace if any hence all of the above conditions hold. X is a pseudocompact space, then cx denotes the usual banach space of continuous scalar valued functions on x.
Recall that a banach lattice is a banac h space endow ed with additional order and. Observations on the separable quotient problem for banach spaces. Smoothness and its equivalents in weakly compactly. Zizler,smoothness and its equivalents in weakly compactly generated banach spaces, to appear in j. The notation we use is standard, with x typically denoting a banach space. In 3 corson asked if, in the context of real or complex banach spaces e, weakly compactly generated banach spaces are exactly those e that are weakly lindel of, i. The banach spacex isweakly compactly generatedwcg if there exists a weakly compact subset of x whose linear span is dense in x. Weak sequential compactness and bornological limit. The class of banach spaces with projectional skeletons is strictly larger than the class of plichko spaces i. A characterization of subspaces of weakly compactly generated banach spaces article pdf available in journal of the london mathematical society 692 april 2004 with 74 reads. If x is a weakly compactly generated wcg banach space, then x admits an equivalent csmooth norm. Some authors also say that a k k space is compactly generated, while others reserve that term for a k k space which is also weak hausdorff, meaning that the image of any t.
Weak compactness in banach spaces pubmed central pmc. Although it is harder to map a nonseparable banach space into a c0. Odell, subspaces of l p which embed into l p, compositio math. Examples of wcg spaces are all separable spaces cf. E fwith te contained in a weakly compactly generated subspace of f.
By a copy of c0 in a locally convex space f we mean the image of a linear homeomorphism from c0 into e. Pdf weakly compact sets and smooth norms in banach spaces. Our main purpose here is to show that the question of di piazza and preiss has negative answer in general. Banach spaces with a countably norming markushevich basis.
Rocky mountain journal of mathematics project euclid. Much of the work on the geometry of banach spaces was related to wcg. This generalizes the concept of free banach lattice over a set of generators, and allows us to study the nakano property and the density character of nondegenerate intervals on these spaces, answering some recent questions of b. A banach space possessing a weakly compact subset cf. Some remarks on weakly compactly generated spaces, israel j. What are the special properties of a banach space b which is generated by a weakly compact, or weakly crcompact, subset. On equivalent characterizations of weakly compactly generated banach spaces article pdf available in rocky mountain journal of mathematics 244 december 1994 with 18 reads how we measure reads. Projections in weakly compactly generated banach spaces. Some remarks on weakly compactly generated banach spaces. If a banach space v is weakly kanalytic and the density character. X is said to be weakly compactly generated in short, wcg if x is the closed linear span of a weakly compact subset of x. In this paper we answer this question in the negative.
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