A weak Grothendieck compactness principle for Banach spaces with a symmetric basis

The Grothendieck compactness principle states that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. In Dowling et al. (J Funct Anal 263(5):1378–1381, 2012), an analogue of the Grothendieck compactness principle for the weak topology was used to characterize Banach spaces with the Schur property. Using a different analogue of the Grothendieck compactness principle for the weak topology, a characterization of the Banach spaces with a symmetric basis that are not isomorphic to $\ell ^1$ and do not contain a subspace isomorphic to $c_0$ is given. As a corollary, it is shown that, in the Lorentz space $d(w,1)$ , every weakly compact set is contained in the closed convex hull of the rearrangement invariant hull of a norm null sequence.     

A Grothendieck compactness principle for the Mackey dual topology

Grothendieck [6] proved that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. In a recent paper [3], an analogous result for weak compactness in a Banach space is shown to be equivalent to the Schur property. In this article, we obtain a similar type result in the Mackey dual of a Banach space. A related result for weak^? compactness is also obtained.     

Super weak compactness and uniform Eberlein compacta

We prove that a topological space is uniform Eberlein compact iff it is homeomorphic to a super weakly compact subset C of a Banach space such that the closed convex hull coC of C is super weakly compact. We also show that a Banach space X is super weakly compactly generated iff the dual unit ball B_X* of X^* in its weak star topology is affinely homeomorphic to a super weakly compactly convex subset of a Banach space.     

本刊英文版Vol.32(2016),No.6论文摘要（英文）

正On Proximinality of Convex Sets in Superspaces Li Xin CHENG Zheng Hua LUO Wen ZHANG Ben Tuo ZHENG Abstract In this paper,we show that a closed convex subset C of a Banach space is strongly proximinal(proximinal,resp.)in every Banach space isometrically containing it if and only if C is locally(weakly,resp.)compact.As a consequence,it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it,the metric     

Dentability and extreme points in Banach spaces

It is shown that a Banach space E has the Radon-Nikodym property (equivalently, every bounded subset of E is dentable) if and only if every bounded closed convex subset of E is the closed convex hull of its strongly exposed points. Using recent work of Namioka, some analogous results are obtained concerning weak¹ strongly exposed points of weak¹ compact convex subsets of certain dual Banach spaces.     

On proximinality of convex sets in superspaces

In this paper, we show that a closed convex subset C of a Banach space is strongly proximinal (proximinal, resp.) in every Banach space isometrically containing it if and only if C is locally (weakly, resp.) compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonempty set-valued and upper semi-continuous.     

A Gelfand-Phillips Property with Respect to the Weak Topology

We consider a Gelfand-Phillips type property for the weak topology. The main results that we obtain are (1) for certain Banach spaces, E^?_? F inherits this property from E and F, and (2) the spaces L^p(μ, E) have this property when E does. A subset A of a Banach space E is a limited set if every (bounded linear) operator T:E → c₀ maps A onto a relatively compact subset of c₀. The Banach space E has the Gelfand-Phillips property if every limited set is relatively compact. In this note, we study the analogous notions set in the weak topology. Thus we say that A ? E is a Grothendieck set if every T: E → c₀ maps A onto a relatively weakly compact set; and E is said to have the weak type GP property if every Grothendieck set in E is relatively weakly compact. In the papers [3, 4 and 6], it is shown among other results that the ?-tensor product E and the spaces L^p(μ, E) inherit the Gelfand-Phillips property from E and F. In this paper, we study the same questions for the weak type GP property. It is easily verified that continuous linear images of Grothendieck sets are Grothendieck and that the weak type GP property is inherited by subspaces. Among the spaces with the weak type GP property one easily finds the separable spaces, and more generally, spaces with a weak* sequentially compact dual ball. Also, C(K) spaces where K is (DCSC) are weak type GP (see [3] and the discussion before Corollary 4 below). A Grothendieck space (a Banach space whose unit ball is a Grothendieck set) has the weak type GP if and only if it is reflexive.     

Convergence of minimizing sequences

A functional f defined on a closed convex subset C of a normed space is to be minimized. It is known that if f is strictly convex and C is compact, then any minimizing sequence converges in norm to a unique minimum. A characterization is given herein for the norm convergence of any minimizing sequence when C is weakly compact and f is strictly quasi-convex, a more general result than those which are already known.     

On extreme points in separable conjugate spaces

It is proved that every bounded closed and convex subset of an arbitrary conjugate separable Banach space is the closed convex hull of its extreme points.     

Distortion and stability of the fixed point property for non-expansive mappings

Let X be a Banach space. We say that X satisfies the fixed point property (weak fixed point property) if every non-expansive mapping defined from a convex closed bounded (convex weakly compact) subset of X into itself has a fixed point. We say that X satisfies the stable fixed point property (stable weak fixed point property) if the same is true for every equivalent norm which is close enough to the original one. Denote by P(X) the set formed by all equivalent norms with the topology of the uniform convergence on the unit ball of X. We prove that the subset of P(X) formed by the norms failing the fixed point property is dense in P(X) when X is a non-distortable space which fails the fixed point property. In particular, no renorming of ?₁ can satisfy the stable fixed point property. Furthermore, we show some examples of distortable spaces failing the weak fixed point property, which can be renormed to satisfy the stable weak fixed point property. As a consequence we prove that every separable Banach space can be renormed to satisfy the stable weak fixed point property.     

Boundaries for Banach spaces determine weak compactness

A boundary for a real Banach space is a subset of the dual unit sphere with the property that each element of the Banach space attains its norm on an element of that subset. Trivially, the pointwise convergence with respect to such a boundary is coarser than the weak topology on the Banach space. The boundary problem asks whether nevertheless both topologies have the same norm bounded compact sets.The main theorem of this paper states the equivalence of countable and sequential compactness of norm bounded sets with respect to an appropriate topology. This result contains, as a special case, the positive answer to the boundary problem and it carries James’ sup-characterization as a corollary.     

A fixed point result in strictly convex Banach spaces

We define an alternate convexically nonexpansive map T on a bounded, closed, convex subset C of a Banach space X and prove that if X is a strictly convex Banach space and C is a nonempty weakly compact convex subset of X, then every alternate convexically nonexpansive map T : C → C has a fixed point. As its application, we give an existence result for the solution of an integral equation.     

The approximation property in terms of the approximability of weak∗-weak continuous operators

By a well-known result of Grothendieck, a Banach space X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous compact operator T:X∗→Y can be uniformly approximated by finite rank operators from X⊗Y. We prove the following “metric” version of this criterion: X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous weakly compact operator T:X∗→Y can be approximated in the strong operator topology by operators of norm ?‖T‖ from X⊗Y. As application, easier alternative proofs are given for recent criteria of approximation property due to Lima, Nygaard and Oja.     

A characterization of reflexive spaces

We prove that a Banach space is reflexive if for every equivalent norm, the set of norm attaining functionals has non-empty norm-interior in the dual space. It is also proved that the set of norm attaining functionals on a Banach space that is not a Grothendieck space is not a w*-G _δ subset of the dual space.     

Un résultat d'existence de sélection mesurable et compacité forte de l'intégrale d'une multiapplication dans un Banach non nécessairement séparable

Summary An existence result of strongly measurable selection of a weakly measurable multifunction from a compact space with non empty convex weakly compact values in a Banach space is given. Some compactness results concerning the set of integrable selections of a weakly measurable multifunction with non empty convex weakly compact values in a weakly compact set of a Banach space and the multivalued integral of such multifunction are presented.     

The point of continuity property in Banach spaces not containing ℓ<Subscript>1</Subscript>

We obtain a local characterization of the point of continuity property for bounded subsets in Banach spaces not containing basic sequences equivalent to the standard basis of ℓ₁ and, as a consequence, we deduce that, in Banach spaces with a separable dual, every closed, bounded, convex and nonempty subset failing the point of continuity property contains a further subset which can be seen inside the set of Borel regular probability measures on the Cantor set in a weak-star dense way. Also, we characterize in terms of trees the point of continuity property of Banach spaces not containing ℓ₁, by proving that a Banach space not containing ℓ₁ satis- fies the point of continuity property if, and only if, every seminormalized weakly null tree has a boundedly complete branch.     

Every weakly compact set can be uniformly embedded into a reflexive Banach space

Based on an application of the Davis-Figiel-Johnson-Pelzyski procedure, this note shows that every weakly compact subset of a Banach space can be uniformly embedded into a reflexive Banach space. As its application, we present the recent renorming theorems for reflexive spaces of Odell- Schlumprecht and Hajek-Johanis can be extended and localized to weakly compact convex subsets of an arbitrary Banach space.     

Farthest points in weakly compact sets

LetS be a weakly compact subset of a Banach spaceB. We show that of all points inB which have farthest points inS contains a denseG ₅ ofB. Also, we give a necessary and sufficient condition for bounded closed convex sets to be the closed convex hull of their farthest points in reflexive Banach spaces.     

Convex-compact sets and Banach discs

Every relatively convex-compact convex subset of a locally convex space is contained in a Banach disc. Moreover, an upper bound for the class of sets which are contained in a Banach disc is presented. If the topological dual E′ of a locally convex space E is the σ(E′,E)-closure of the union of countably many σ(E′,E)-relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact.     

局部Cauchy列和局部完备性的性质

利用完备化的方法,给出了局部凸分离空间（X,T）中序列{xn}是局部Cauchy列当且仅当存在单调增且趋于正无穷大的正实数列{an},使得min{an,am}（xn-xm）→0（m,n→∞）,并得到局部凸分离空间（X,T）是局部完备的当且仅当X中每个丁局部Cauchy列的绝对凸闭包是丁紧的,以及一些局部完备性的相关性质．