On Lipschitz-free spaces over spheres of Banach spaces (original) (raw)
A note on ball-covering property of Banach spaces
Journal of Mathematical Analysis and Applications, 2010
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 S X 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 S X 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.
Covering spheres of Banach spaces by balls
Mathematische Annalen, 2009
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.
A New Class of Banach Spaces and Its Relation with Some Geometric Properties of Banach Spaces
Abstract and Applied Analysis, 2012
By introducing the concept ofL-limited sets and thenL-limited Banach spaces, we obtain some characterizations of it with respect to some well-known geometric properties of Banach spaces, such as Grothendieck property, Gelfand-Phillips property, and reciprocal Dunford-Pettis property. Some complementability of operators on such Banach spaces are also investigated.
NOTE ON A PROPERTY OF THE BANACH SPACES
We show that we may consider a partial ordering ≤ in an infinite dimensional Banach space ( kk), which we obtain through any normed Hamel base of the space, such that ( kk ≤) is a Banach lattice.
An isomorphic characterization of L¹-spaces
Indagationes Mathematicae, 2007
We show that a sequentially (τ)-complete topological vector lattice X_τ is isomorphic to some L¹(μ), if and only if the positive cone can be written as X₊ = R₊·B for some convex, (τ)-bounded, and (τ)-closed set B ⊂ X₊\{0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness (monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the "ball-generated" ordering induced by the cone Y₊ = R₊· B̅(u,1) (for ‖u‖ > 1) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials.
Lipschitz functions on spaces of homogeneous type
Advances in Mathematics, 1979
Lipschitz Functions on Spaces of Homogeneous Type Results on the geometric structure of spaces of homogeneous type are obtained and applied to show the equivalence of certain classes of Lipschitz functions defined on these spaces. I. YOTATION AND DEFINITIONS By a quasi-distance on a set X we mean a non-negative function d(x, y) defined on S x X, such that (i) for every x' and y in S, d(s, y) = 0 if and only if s = y, (ii) for every .v and y in X, d(~, y) = d(y, X) and (iii) there exists a finite constant K such that for every X, y and z in S d(x, y) ,(q+, 2) + d(z, y)). A quasi-distance d(x, y) defines a uniform structure on X. The balls B(x, r) = {y: 4% Y) < 4, Y > 0, form a basis of neighbourhoods of w for the topology induced by the uniformity on ,Y. This topology is a metric one since the uniform structure associated to d(~, y) has a countable basis. We shall refer to this topology as the d-topology of X. We say that two quasi-distances d(x, y) and d'(~, y) on S are equivalent if there exist two positive and finite constants, c1 and ca , such that c&x, y) < d'(~, y) < c&(x, y) hold for every x and y in S. We observe that the uniformities and the topologies defined by equivalent quasidistances coincide. Let X be a set endowed with a quasi-distance d(.v, y) and assume that a positive measure CL, defined on a a-algebra of subsets of X which contains the d-open subsets and the balls B(x, Y), is given and satisfies that there exist two finite constants, a > 1 and A, such that 0 < p(B(x, UT)) < A .