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.     

Finitely locally finite coverings of Banach spaces

A well-known result due to H. Corson states that, for any covering τ by closed bounded convex subsets of any Banach space X containing an infinite-dimensional reflexive subspace, there exists a compact subset C of X that meets infinitely many members of τ. We strengthen this result proving that, even under the weaker assumption that X contains an infinite-dimensional separable dual space, an (algebraically) finite-dimensional compact set C with that property can always be found.     

Banach spaces of universal disposition

In this paper we present a method to obtain Banach spaces of universal and almost-universal disposition with respect to a given class M of normed spaces. The method produces, among others, the only separable Banach space of almost-universal disposition with respect to the class F of finite-dimensional spaces (Gurari? space G); or the only, under CH, Banach space with density character the continuum which is of universal disposition with respect to the class S of separable spaces (Kubis space K). We moreover show that K is isomorphic to an ultrapower of the Gurari? space and that it is not isomorphic to a complemented subspace of any C(K)-space. Other properties of spaces of universal disposition are also studied: separable injectivity, partially automorphic character and uniqueness.     

An approximation problem of the finite rank operator in Banach spaces

By the method of geometry of Banach spaces, we have proven that a bounded linear operator in Banach space is a compact linear one iff it can be uniformly approximated by a sequence of the finite rank bounded homogeneous operators, which reveals the essence of the counter example given by Enflo.     

Uniformly smooth renorming of Banach spaces with modulus of convexity of power type 2

An upper bound q(c) for the best, under equivalent renorming, possible power type of the modulus of smoothness of a Banach space with modulus of convexity satisfying δX(ε)?cε², is found. The estimate is asymptotically sharp and is expressed in terms of linear fractional function q(c).     

k-super-strongly convex and k-super-strongly smooth Banach spaces

In this paper, we introduce two types of new Banach spaces: k-super-strongly convex spaces and k-super-strongly smooth spaces. It is proved that these two notions are dual. We also prove that the class of k-super-strongly convexifiable spaces is strictly between locally k-uniformly rotund spaces and k-strongly convex spaces, and obtain some necessary and sufficient conditions of k-super-strongly convex space (respectively k-super-strongly smooth space). Also, for each k?2, it is shown that there exists a k-super-strongly convex (respectively k-super-strongly smooth) space which is not (k−1)-super-strongly convex (respectively (k−1)-super-strongly smooth) space.     

On uniform convexity of Banach spaces

This paper gives some relations and properties of several kinds of generalized convexity in Banach spaces. As a result, it proves that every kind of uniform convexity implies the Banach-Sakes property, and several notions of uniform convexity in literature are actually equivalent.     

Constructive utility functions on Banach spaces

In this paper we prove the existence of continuous order-preserving functions on subsets of ordered Banach spaces using a constructive approach.     

Geometric mean and triangles inscribed in a semicircle in Banach spaces

We consider the triangles with vertices x, −x and y where x,y are points on the unit sphere of a normed space. Using the geometric means of the variable lengths of the sides of these triangles, we define two geometric constants for Banach spaces. These constants are closely related to the modulus of convexity of the space under consideration, and they seem to represent a useful tool to estimate the exact values of the James and Jordan–von Neumann constants of some Banach spaces.     

The uniform convexity of the Edmunds-Triebel logarithmic spaces

We show how uniform convexity can be preserved in the logarithmic spaces A_θ(logA)_b,p. Estimates are given for the moduli of convexity of A_θ(logA)_b,p in terms of the moduli of A₀ and A₁, when one or both of them are uniformly convex.     

A class of primary Banach spaces

It is shown that if X is an infinite-dimensional Banach space with a boundedly complete subsymmetric basis, then the infinite l^∞ direct sum l^∞(X⊕X⊕X⊕?) is primary.     

关于Banach空间的强凸性

本文讨论强凸性、Ｌ－ｋＲ，ＬωＲ和（Ｇ）性质之间的关系，指出强凸性介于ＬωＲ和（Ｇ）性质之间，证明光滑的有（Ｇ）性质的Ｂａｎａｃｈ空间是强凸的，此外指出存在一个Ｂａｎａｃｈ空间Ｘ，它是ＬωＲ但对任意自然数ｋ，Ｘ不是Ｌ－ｋＲ．     

Some Characterizations of Hereditarily Indecomposable Banach Spaces

In this paper we first discuss the relations between some G-M-type spaces, and the previous eight kinds of G-M-type Banach spaces are merged into four different kinds. Then we build a Generalized Operator Extension Theorem, and introduce the concept of complete minimal sequences. Some sufficient and necessary conditions under which a Banach space is a hereditarily indecomposable space are given. Finally, we give some characterizations of hereditarily indecomposable Banach Spaces.     

具有AUMD性质的复Banach空间的某些特征

本应用解析鞅的一类特殊的不等式给出了具有AUMD性质的复Banach空间的某些特征。     

Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

This paper studies the convergence of the sequence defined by x0∈C,xn 1=αnu (1-αn)Txn,n=0,1,2,…, where 0 ≤αn ≤ 1, limn→∞αn = 0, ∑∞n=0 αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.     

On K_0-groups of operator algebras on Banach spaces

This paper merges some classifications of G-M-type Banach spaces simplifically, discusses the condition of K0(B(X)) = 0 for operator algebra B(X) on a Banach space X, and obtains a result to improve Laustsen's sufficient condition, gives an example to show that X ≈ X2 is not a sufficient condition of K0(B(X)) = 0.     

Colocality and twisted sums of Banach spaces

Using the relation between subspaces of Banach spaces and quotients of their duals, we introduce the concept of colocality to give a new method that guarantees the existence of nontrivial twisted sums in which finite quotients play a major role (Theorem 1.7). An interesting point is that no restrictions are imposed on the quotients, only on the various subspaces. New examples of nontrivial twisted sums are given.     

Best approximation in polyhedral Banach spaces

In the present paper, we study conditions under which the metric projection of a polyhedral Banach space X onto a closed subspace is Hausdorff lower or upper semicontinuous. For example, we prove that if X satisfies (∗) (a geometric property stronger than polyhedrality) and Y⊂X is any proximinal subspace, then the metric projection P_Y is Hausdorff continuous and Y is strongly proximinal (i.e., if {y_n}⊂Y, x∈X and , then ).One of the main results of a different nature is the following: if X satisfies (∗) and Y⊂X is a closed subspace of finite codimension, then the following conditions are equivalent: (a) Y is strongly proximinal; (b) Y is proximinal; (c) each element of Y^⊥ attains its norm. Moreover, in this case the quotient X/Y is polyhedral.The final part of the paper contains examples illustrating the importance of some hypotheses in our main results.