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.     

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.     

Ball-covering property of Banach spaces that is not preserved under linear isomorphisms

By a ball-covering B of a Banach space X, we mean that it is a collection of open balls off the origin whose union contains the sphere of the unit ball of X. The space X is said to have a ball-covering property, if it admits a ball-covering consisting of countably many balls. This paper, by constructing the equivalent norms on l~∞, shows that ball-covering property is not invariant under isomorphic mappings, though it is preserved under such mappings if X is a Gateaux differentiability space; presents that this property of X is not heritable by its closed subspaces; and the property is also not preserved under quotient mappings.     

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.     

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.     

Espaces de Banach superstables,distances stables et homeomorphismes uniformes

We introduce here the notion of superstable Banach space, as the superproperty associated with the stability property of J. L. Krivine and B. Maurey. IfE is superstable, so are theL ^p (E) for eachp∈[1, +∞[. If the Banach spaceX uniformly imbeds into a superstable Banach space, then there exists an equivalent invariant superstable distance onX; as a consequenceX contains subspaces isomorphic tol ^p spaces (for somep∈[1, ∞[). We give also a generalization of a result of P. Enflo: the unit ball ofc ₀ does not uniformly imbed into any stable Banach space.     

On the heredity of weak compact generating

IfX is a Banach space such thatX, X* are subspaces of Banach spaces generated by weakly-compact sets, thenX is also generated by a weakly-compact set and admits an equivalent Fréchet smooth norm.     

Norming sets and integration with respect to vector measures

Let v be a countably additive measure defined on a measurable space (Ω, Σ) and taking values in a Banach space X. Let f : Ω → ? be a measurable function. In order to check the integrability (respectively, weak integrability) of f with respect to v it is sometimes enough to test on a norming set Λ ⊂ X^*. In this paper we show that this is the case when A is a James boundary for BX^* (respectively, Λ is weak^*-thick). Some examples and applications are given as well.     

Existence of best approximation elements inC(Q,X)

Generalizing the result of A. L. Garkavi (the caseX = ?) and his own previous result concerningX = ?), the author characterizes the existence subspaces of finite codimension in the spaceC(Q, X) of continuous functions on a bicompact spaceQ with values in a Banach spaceX, under some assumptions concerningX. Under the same assumptions, it is proved that in the space of uniform limits of simple functions, each subspace of the form $$\left\{ {g \in B:\smallint _Q \left\langle {g(t),d\mu _i } \right\rangle = 0,i = 1,...,n} \right\},$$ whereμ _i ∈ C(Q, X)^* are vector measures of regular bounded variation, is an existence subspace (the integral is understood in the sense of Gavurin).     

Affine functions on simplexes and extreme operators

IfK a simplex andX a Banach space thenA(K, X) denotes the space of affine continuous functions fromK toX with the supremum norm. The extreme points of the closed unit ball ofA(K, X) are characterized,X being supposed to satisfy certain conditions. This characterization is used to investigate the extreme compact operators from a Banach spaceX to the spaceA(K)=A(K, (− ∞, ∞)). This note is part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Prof. A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank them for their helpful advice and kind encouragement.     

The space of maps from a locally compact space to a Banach space

The space of continuous maps from a topological spaceX to topological spaceY is denoted byC(X,Y) with the compact-open topology. In this paper we prove thatC(X,Y) is an absolute retract ifX is a locally compact separable metric space andY a convex set in a Banach space. From the above fact we know thatC(X,Y) is homomorphic to Hilbert spacel ₂ ifX is a locally compact separable metric space andY a separable Banach space; in particular,C(R ⁿ,R^m) is homomorphic to Hilbert spacel ₂. This research is supported by the Science Foundation of Shanxi Province's Scientific Committee     

Covering spheres of Banach spaces by balls

If the unit sphere of a Banach space X can be covered by countably many balls no one of which contains the origin, then, as an easy consequence of the separation theorem, X* is w*-separable. We prove the converse under suitable renorming. Moreover, the balls of the countable covering can be chosen as translates of the same ball. Research of V. P. Fonf was supported in part by Israel Science Foundation, Grant # 139/02 and by the Istituto Nazionale di Alta Matematica of Italy. Research of C. Zanco was supported in part by the Ministero dell’Università e della Ricerca Scientifica e Tecnologica of Italy and by the Center for Advanced Studies in Mathematics at the Ben-Gurion University of the Negev, Beer-Sheva, Israel.     

The duality between Asplund spaces and spaces with the Radon-Nikodym property

A Banach spaceX is an Asplund space (a strong differentiability space) if and only ifX ^* has the Radon-Nikodym property.     

On almost sure convergence of amarts and martingales without the Radon-Nikodym property

It is shown here that for any Banach spaceE-valued amart (X _n) of classB, almost sure convergence off(X_n) tof(X) for eachf in a total subset ofE ^* implies scalar convergence toX.     

空间的几种超投影性质及其局部化

讨论Banach空间几种超投影性质(及其相应的局部化性质)之间的关系,证明了在Banach空间X自反的条件下,X是l_p-次投影空间的充要条件是X^*是l_p-超投影空间,X是局部l_p-次投影空间的充要条件是X^*是局部l_p-超投影空间,以及X是局部次投影空间的充要条件是X^*是局部超投影的。其中1/p+1/q=1(p>1,q>1)。     

Fixed point theory and product,sub-, super-spaces

In this paper, we definen-segmentwise metric spaces and then we prove the following results:

1. (i)|Let (X, d) be ann-segmentwise metric space. ThenX ⁿ has the fixed point property with respect to uniformly continuous bounded functions if and only if, for any continuous functionF: C ^*(X) → C^*(X) and for anyn-tuple of distinct points x₁, x₂, ?, x_n ∈X, there exists anh ∈C ^*(X) such that $$F(h)(x_1 ) = h(x_1 ),i = 1,2,...,n;$$ whereC ^*(X) has either the uniform topology or the subspace product (Tychonoff) topology \((C^ * (X) \subseteq X^X )\) .
2. LetX _i (i = 1, 2, ?) be countably compact Hausdorff spaces such thatX ₁ × ? × X_n has the fixed point property for alln ∈N Then the product spaceX ₁ × X₂ × ? has the fixed point property. We shall also discuss several problems in the Fixed Point Theory and give examples if necessary. Among these examples, we have:
3. There exists a connected metric spaceX which can be decomposed as a disjoint union of a closed setA and an open setB such thatA andB have the fixed point property andX does not have.
4. There exists a locally compact metrizable spaceX which has the fixed point property but its one-point compactificationX ⁺ does not have the fixed point property.

Other relevant results and examples will be presented in this paper.     

w* sequential convergence

LetX be an infinite dimensional Banach space, andX* its dual space. Sequences {χ _n ^* } _n=1 ^∞ ?X* which arew* converging to 0 while inf _n ‖x* _n ‖>0, are constructed.     

Extreme points and retractions in Banach spaces

ForT a completely regular topological space andX a strictly convex Banach space, we study the extremal structure of the unit ball of the spaceC(T,X) of continuous and bounded functions fromT intoX. We show that when dimX is an even integer then every point in the unit ball ofC(T, X) can be expressed as the average of three extreme points if, and only if, dimT< dimX, where dimT is the covering dimension ofT. We also prove that, ifX is infinite-dimensional, the aforementioned representation of the points in the unit ball ofC(T, X) is always possible without restrictions on the topological spaceT. Finally, we deduce from the above result that the identity mapping on the unit ball of an infinite-dimensional strictly convex Banach space admits a representation as the mean of three retractions of the unit ball onto the unit sphere. The author wishes to express his gratitude to Dr. Juan Francisco Mena Jurado for many helpful suggestions during the preparation of this paper.     

Characterization and properties of extreme operators intoC(Y)

LetE be a real (or complex) Banach space,Y a compact Hausdorff space, andC(Y) the space of real (or complex) valued continuous functions onY. IfT is an extreme point in the unit ball of bounded linear operators fromE intoC(Y), then it is shown thatT ^* maps (the natural imbedding inC(Y) ^* of)Y into the weak ^*-closure of extS(E ^*), provided thatY is extremally disconnected, orE=C(X), whereX is a dispersed compact Hausdorff space.     

A complementary universal conjugate Banach space and its relation to the approximation problem

LetC ₁=(ΣG _n ) _{l 1}, where (G _n ) is a sequence which is dense (in the Banach-Mazur sense) in the class of all finite dimensional Banach spaces. IfX is a separable Banach space, thenX ^* is isometric to a subspace ofC ₁ ^* =(ΣG _n ^* ) _m which is the range of a contractive projection onC ₁ ^* . Separable Banach spaces whose conjugates are isomorphic toC ₁ ^* are classified as those spaces which contain complemented copies of C₁. Applications are that every Banach space has the [metric] approximation property ([m.] a.p., in short) iff (ΣG _n ^* ) _m does, and if there is a space failing the m.a.p., thenC ₁ can be equivalently normed to fail the m.a.p. The author was supported in part by NSF GP 28719.