Intuitively, the center of a line segment inside the unit ball must lie deep inside the unit ball unless the segment is short properties. Eudml a uniformly convex banach space which contains no. A short course on non linear geometry of banach spaces 3 we nish this very short section by mentioning an important recent result by g. Metric characterizations of some classes of banach spaces mikhail ostrovskii november 22, 2014 dedicated to the memory of cora sadosky abstract.
Saejung proved that fconvex banach spaces have uniform normal structure. Characteristic inequalities of uniformly convex and. In the theory of banach space geometry, geometric constants which quantify various geometric features of banach spaces often play fundamental roles. More structure on k affects the properties of the banach spaceck as shown in the following theorem. Geometric mean and triangles inscribed in a semicircle in. Definitions of locally uniform convergence in banach spaces. Banach spaces, james constant, jordanvon neumann constant, uniformly nonsquare spaces, nth james constants, bconvex spaces, khintchine constants, type. Heinz means and triangles inscribed in a semicircle in banach spaces author.
Pdf geometric constants of banach spaces and the matrix. Moreover, we construct an equivalent almost square. Metric characterizations of some classes of banach spaces. Title uniform convexity, uniform nonsquareness and. Uniform nonsquareness and uniform normal structure in banach spaces ji gao communicated by hari berovicci let xbe a banach space, and bx and sx be the unit ball and unit sphere of x. A uniformly convex space is a normed vector space such that, for every such that for any two vectors with. Also, it is well known that there exists a banach space which is neither uniformly convex nor uniformly nonsquare but surperrefrexive. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.
On some geometric constants in banach spaces springerlink. In the geometric theory of banach spaces, the concept of uniform convexity plays a very significant. X of nearly uniform smoothness of a reflexive banach space satisfies. As special cases for, two necessary and sufficient conditions of uniform extreme convexity are presented and a characteristic inequality of uniformly extremely convex spaces is given 1. Complete metric spaces notions such as convergent sequence and cauchy sequence make sense for any metric space. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. On nth james and khintchine constants of banach spaces.
The hahnbanach theorem is nontrivial, but does not use completeness. Uniformly nonsquare banach spaces have the fixed point. Uniformly convex and uniformly smooth banach spaces. Using this general result, we find sufficient conditions in order that. In this paper, we first introduce a family of geometric constants of a real normed space x and give some results concerning these constants. Uniformly nonsquare banach spaces have the fixed point property. Uniformly normal structure and related coefficients. Uniformly nonsquare points and representation of functionals of orliczbochner sequence spaces shi, zhongrui and wang, yu, rocky mountain journal of mathematics, 2018 minimal homeomorphisms and approximate conjugacy in measure lin, huaxin, illinois journal of mathematics, 2007. Since and uniform limits of continuous functions are continuous, then ck is a closed subspace of bk and hence a banach space. A normed linear space b is isometric to a subspace of its second conjugate space under the. Pdf on jan 1, 1997, mikio kato and others published some characterizations of uniformly nonsquare banach spaces find, read and cite all the research you need on researchgate.
Then, we give some characterizations of hilbert spaces and uniformly nonsquare spaces and obtain sufficient conditions for normal structure related to. Let x be a banach space, and let sx denote the unit sphere of x. A schauder basis in a banach space x is a sequence e n n. It is shown that if the modulus x of nearly uniform smoothness of a reflexive banach space satisfies x. In this paper, we investigate the locally uniformly non.
A banach space x is called uniformly nonsquare james, 1964 if there exists a. Altshuler, a banach space with a symmetric basis which contains no. Ii x has the weak fixed point property for nonexpansive mappings whenever ra, x 0. This paper is mainly concerned with, in particular, one of the bestknown geometric constants of banach spaces, that is, james constant. Finally, as made clear in work of gelfand, of grothendieck, and of many others, many subtler sorts of topological vectorspaces are expressible as limits of banach spaces, making clear that banach spaces play. Browse other questions tagged functionalanalysis banachspaces or ask your own question.
Throughout, f will denote either the real line r or the complex plane c. A banach space x is said to have norinal structure 2, 8 if. Godefroykalton 2003 let xand ybe separable banach spaces and suppose that f. Uniformly non square banach spaces it is known 6, th. A nonreflexive banach space that is uniformly nonoctahedral. Heinz means and triangles inscribed in a semicircle in. Pdf we show that finite dimensional banach spaces fail to be uniformly non locally almost square. Banach spaces these notes provide an introduction to banach spaces, which are complete normed vector spaces.
As of now k can be any haussdorf topological space. The main purpose of the paper is to present some recent results on metric characterizations of superre exivity and the radonnikodym property. Ballcovering property in uniformly non banach spaces and. Stated in geometric terms, a norm is uniformly convex if whenever the midpoint of a variable chord in the unit sphere of the space approaches the boundary of the. Beck, a convexity condition in banach spaces and the strong law of large numbers, proc. Banach spaces with a schauder basis are necessarily separable, because the countable set of finite linear combinations with rational coefficients say is dense.
Uniformly nonsquare banach spaces it is known 6, th. These two spaces, as well as the other ones discussed below. Both classes of spaces are uniformly nonsquare, their properties are being studied. On two classes of banach spaces with uniform normal structure.