Some geometric and topological properties of Banach spaces via ball coverings

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere of X; and X is said to have the ball-covering property provided it admits a ball-covering of countably many balls. This paper shows that Gδ property of points in a Banach space X endowed with the ball topology is equivalent to the space X admitting the ball-covering property. Moreover, smoothness, uniform smoothness of X can be characterized by properties of ball-coverings of its finite dimensional subspaces.     

Characterizations of universal finite representability and b-convexity of Banach spaces via ball coverings

By a ball-covering B of a Banach space X, wemean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere of X; and X is said to have the ball-covering property provided it admits a ball-covering of countably many balls. This paper shows that universal finite representability and B-convexity of X can be characterized by properties of ball-coverings of its finite dimensional subspaces.     

A note on ball-covering property of Banach spaces

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere SX of X; and X is said to have the ball-covering property (BCP) provided it admits a ball-covering by countably many balls. In this note we give a natural example showing that the ball-covering property of a Banach space is not inherited by its subspaces; and we present a sharp quantitative version of the recent Fonf and Zanco renorming result saying that if the dual X^∗ of X is w^∗ separable, then for every ε>0 there exist a (1+ε)-equivalent norm on X, and an R>0 such that in this new norm SX admits a ball-covering by countably many balls of radius R. Namely, we show that R=R(ε) can be taken arbitrarily close to (1+ε)/ε, and that for X=?₁[0,1] the corresponding R cannot be equal to 1/ε. This gives the sharp order of magnitude for R(ε) as ε→0.     

Ball-covering property of Banach spaces

We consider the following question: For a Banach spaceX, how many closed balls not containing the origin can cover the sphere of the unit ball? This paper shows that: (1) IfX is smooth and with dimX=n<∞, in particular,X=R ⁿ,then the sphere can be covered byn+1 balls andn+1 is the smallest number of balls forming such a covering. (2) Let Λ be the set of all numbersr>0 satisfying: the unit sphere of every Banach spaceX admitting a ball-covering consisting of countably many balls not containing the origin with radii at mostr impliesX is separable. Then the exact upper bound of Λ is 1 and it cannot be attained. (3) IfX is a Gateaux differentiability space or a locally uniformly convex space, then the unit sphere admits such a countable ball-covering if and only ifX ^* isw ^*-separable.     

Every Banach space with a <Emphasis Type="Italic">w</Emphasis><Superscript>*</Superscript>-separable dual has a 1+<Emphasis Type="Italic">ɛ</Emphasis>-equivalent norm with the ball covering property

A normed space is said to have ball-covering property if its unit sphere can be contained in the union of countably many open balls off the origin. This paper shows that for every ɛ > 0 every Banach space with a w ^*-separable dual has a 1+ɛ-equivalent norm with the ball covering property.     

Weak continuity of Riemann integrable functions in Lebesgue-Bochner spaces

In general, Banach space-valued Riemann integrable functions defined on [0, 1] （equipped with the Lebesgue measure） need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space LX^1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX^1 has the weak Lebesgue property whenever X has the Radon-Nikodym property and X＊ is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31（2）, 697-703 （2001）] that L^1[0, 1] has the weak Lebesgue property.     

Tilings,packings, coverings,and the approximation of functions

A packing (resp. covering) ? of a normed space X consisting of unit balls is called completely saturated (resp. completely reduced) if no finite set of its members can be replaced by a more numerous (resp. less numerous) set of unit balls of X without losing the packing property (resp. covering property) of ?. We show that a normed space X admits completely saturated packings with disjoint closed unit balls as well as completely reduced coverings with open unit balls, provided that there exists a tiling of X with unit balls. Completely reduced coverings by open balls are of interest in the context of an approximation theory for continuous real‐valued functions that rests on so‐called controllable coverings of compact metric spaces. The close relation between controllable coverings and completely reduced coverings allows an extension of the approximation theory to non‐compact spaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

广义A-proper映射的拓扑度

本文提出了广义A-proper映射,它更弱于A-proper映射。通过建立广义A-proper度,可用来研究(M)型映射与部分解答Browder问题(见[4];用类似[6]的方法)。本文推广了[3]、[6]和[2]的工作。 今往设X、y为实Banach空间,D X是开集。D表其边界,是之闭包。“→”和“”分别表强和弱收敛。N为正整数集合。     

On paratopological vector spaces

We show that each first countable paratopological vector space X has a compatible translation invariant quasi-metric such that the open balls are convex whenever X is a pseudoconvex vector space. We introduce the notions of a right-bounded subset and of a right-precompact subset of a paratopological vector space X and prove that X is quasi-normable if and only if the origin has a convex and right-bounded neighborhood. Duality in this context is also discussed. Furthermore, it is shown that the bicompletion of any paratopological vector space (respectively, of any quasi-metric vector space) admits the structure of a paratopological vector space (respectively, of a quasi-metric vector space). Finally, paratopological vector spaces of finite dimension are considered. This revised version was published online in June 2006 with corrections to the Cover Date.     

Locally Compact Path Spaces

It is shown that the space X^[0,1], of continuous maps [0,1]X with the compact-open topology, is not locally compact for any space X having a nonconstant path of closed points. For a T₁-space X, it follows that X^[0,1] is locally compact if and only if X is locally compact and totally path-disconnected. Mathematics Subject Classifications (2000) 54C35, 54E45, 55P35, 18B30, 18D15.     

Norm-to-Weak Upper Semi-Continuity of the Duality and Pre-Duality Mappings

We show that when the duality map is norm-to-weak upper semi-continuous at some point of a dual space, the pre-duality map shares this property. We show that if x is a point of very smoothness of a Banach space X, it fails in general to be a point of very smoothness of the bidual X ^**. This cannot happen however if the bidual X ^** is a Grothendieck space.     

The Priestley Separation Axiom for Scattered Spaces

Let R be a quasi-order on a compact Hausdorff topological space X. We prove that if X is scattered, then R satisfies the Priestley separation axiom if and only if R is closed in the product space X×X. Furthermore, if X is not scattered, then we show that there is a quasi-order on X that is closed in X×X but does not satisfy the Priestley separation axiom. As a result, we obtain a new characterization of scattered compact Hausdorff spaces.     

On the super fixed point property in product spaces

We prove that if F is a finite-dimensional Banach space and X has the super fixed point property for nonexpansive mappings, then F⊕X has the super fixed point property with respect to a large class of norms including all lp norms, 1?p<∞. This provides a solution to the “super-version” of the problem of Khamsi (1989).     

Web‐compact spaces,Fréchet‐Urysohn groups and a Suslin closed graph theorem

A topological space X is strongly web‐compact if X admits a family {A_α: α ∈ ?^?} of relatively countably compact sets covering X and such that A_α ? A_β for α ≤ β. The main result of this paper states the following: Theorem A Let X and Y be topological groups and f a homomorphism between X and Y with closed graph. If X is Fréchet‐Urysohn and Baire and Y is strongly web‐compact, then f is continuous. This extends a result of Valdivia. We provide an example showing that the property of being strongly web‐compact is not productive. This applies to show that there are quasi‐Suslin spaces X whose product X × X is not quasi‐Suslin (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On smooth, nonlinear surjections of banach spaces

It is shown that (1) every infinite-dimensional Banach space admits aC ¹ Lipschitz map onto any separable Banach space, and (2) if the dual of a separable Banach spaceX contains a normalized, weakly null Banach-Saks sequence, thenX admits aC ^∞ map onto any separable Banach space. Subsequently, we generalize these results to mappings onto larger target spaces. Supported by an NSF Postdoctoral Fellowship in Mathematics.     

A glance at spaces with closure-preserving local bases

Call a space X (weakly) Japanese at a pointx∈X if X has a closure-preserving local base (or quasi-base respectively) at the point x. The space X is (weakly) Japanese if it is (weakly) Japanese at every x∈X. We prove, in particular, that any generalized ordered space is Japanese and that the property of being (weakly) Japanese is preserved by σ-products; besides, a dyadic compact space is weakly Japanese if and only if it is metrizable. It turns out that every scattered Corson compact space is Japanese while there exist even Eberlein compact spaces which are not weakly Japanese. We show that a continuous image of a compact first countable space can fail to be weakly Japanese so the (weak) Japanese property is not preserved by perfect maps. Another interesting property of Japanese spaces is their tightness-monolithity, i.e., in every weakly Japanese space X we have for any set A⊂X.     

Images and Open Subspaces of SV Spaces

A topological space is finitely an F-space if its Stone–?ech compactification is a union of finitely many closed F-spaces and a space is SV if C(X) has the property that C(X)/P is a valuation domain for each prime ring ideal P of C(X). This article studies the images under open continuous functions and the open subspaces of spaces that are finitely an F-space or are SV. It is shown that an open continuous image of a compact space that is finitely an F-space is finitely an F-space and an open continuous image of certain SV spaces is SV. Also, it is shown cozerosets, but not necessarily open sets, of SV spaces are SV spaces and a similar situation holds for spaces that are finitely an F-space.     

A full-invariant theorem and some applications

Let (X,τ₁) and (Y,τ₂) be two Hausdorff locally convex spaces with continuous duals X′ and Y′, respectively, L(X,Y) be the space of all continuous linear operators from X into Y, K(X,Y) be the space of all compact operators of L(X,Y). Let WOT and UOT be the weak operator topology and uniform operator topology on K(X,Y), respectively. In this paper, we characterize a full-invariant property of K(X,Y); that is, if the sequence space λ has the signed-weak gliding hump property, then each λ-multiplier WOT-convergent series ∑_iT_i in K(X,Y) must be λ-multiplier convergent with respect to all topologies between WOT and UOT if and only if each continuous linear operator T :(X,τ₁)→(λ^β,σ(λ^β,λ)) is compact. It follows from this result that the converse of Kalton's Orlicz–Pettis theorem is also true.     

Dense single-valuedness of monotone operators

It is shown that the set of points for which a monotone mappingT:X→X ^* from a separable Banach space into its dual is not single-valued has no interior; if dimX<∞ and intD(T)≠ϕ then the set has Lebesgue measure zero. Moreover, for accretive mappingsT:X→X from a separable Banach space into itself, the dimension of the set of points whose images contain balls of codimension not larger thank does not exceedk. Applications to convexity are given.     

Dual problems for weak and quasi approximation properties

It is shown that for the separable dual X^∗ of a Banach space X if X^∗ has the weak approximation property, then X has the metric quasi approximation property. Using this it is shown that for the separable dual X^∗ of a Banach space X the quasi approximation property and metric quasi approximation property are inherited from X^∗ to X and for a separable and reflexive Banach space X, X having the weak approximation property, bounded weak approximation property, quasi approximation property, metric weak approximation property, and metric quasi approximation property are equivalent. Also it is shown that the weak approximation property, bounded weak approximation property, and quasi approximation property are not inherited from a Banach space X to X^∗.