The Mazur property for compact sets

Necessary and sufficient conditions for a Banach space with the Mazur intersection property to be an Asplund space are given. It is proved that Mazur intersection property is determined by the separable subspaces of the space. Corresponding problems for a space to have the ball-generated property are considered. Some comments on possible renorming so that a space having the Mazur intersection property are given.A normed space X is said to have the Mazur's intersection property (MIP) if for every bounded closed convex set K in X, there exists a family of closed balls, {J3i}«€/> such that K -f] Bi. For a summary of the (MIP), we refer to [3]. A set K in X is said to be ball-generated [5] if K = (~j Ki where each Ki is a finite union of closed balls in X. We say that X has the ball-generated property (BGP) if every bounded closed convex set in X is ball-generated.The problem raised in [4] whether every Banach space with the (MIP) is an Asplund space is still open. In this note, we give some necessary and sufficient conditions for a Banach space with (MIP) to be an Asplund space. We show that the (MIP) is determined by separable subspaces of the space. Similar problems for space with the (BGP) are considered. We make some comments on possible renorming for a Banach space to have the (MIP).For a Banach space X, let Bx = {x : x 6 X, \\x\\ ^ 1} and Sx = {x : x E X, \\x\\ = 1}. For a set K in X, let foK be the closed convex hull of K and let [K] be the closed linear subspace in X spanned by K.

mentioning

confidence: 99%

Balls intersection properties of Banach spaces

Chen

Lin

1992

“…Thus h belongs to the cone of extreme points of LP(n, X)*. Therefore LP(n, X) has property (CI) [14] and the proof is complete. 0…”

Section: \(F 9) -T(g)\ < E Nmentioning

confidence: 71%

“…In [4], Giles, Gregory and Sims proved that X has property (I) if and only if the weak* denting points of the unit ball of X* are dense in the unit sphere. In [14], Sersouri proved that X has property (CI) if and only if the cone of extreme points of X* is dense in X* for the topology of uniform convergence on compact sets of X, where the cone of extreme points of X* is the cone generated by the set of extreme points of Bx' • PROPOSITION 9 . Suppose the measure space space (fi, E, /x) is not purely atomic.…”

Section: J) = H-(mo I £ | | ) = Lim (Mo ^T T £^Iimoii •mentioning

confidence: 99%

Asymptotic-norming and Mazur intersection properties in Bochner function spaces

Lin

1993

Unless otherwise stated, we always assume X is a Banach space, 1 < p, q < oo with l / p + I/? = 1, and (fi, S, fj.) is a positive measure space so that E contains an element with finite positive measure. We use Sx and Bx to denote the unit sphere and the unit ball in X respectively.The asymptotic-norming property (ANP) was introduced by James and Ho [12]. There are three different kinds of asymptotic-norming properties, and each of them implies the Radon-Nikodym property [12]. Ghoussoub and Maurey [3] proved that for separable Banach spaces the asymptotic-norming property is equivalent to the RadonNikodym property. However, in general, it is an open question whether the two properties are equivalent.A set $ in X* is a norming set for X if $ C Bx* and $ norms X, that is, ||x|| = sup/(x) for all x in X. A sequence {x n } is asymptotically normed by $ if for /e* each e > 0, there are TV ^ 1 and / € $ such that f(x n ) > ||x n || -e, for all n ^ N. We say that X has the %-ANP-I (respectively S-ANP-II; or S-ANP-M) if every sequence {x n } in Sx that is asymptotically normed by $ is convergent (respectively has a convergent subsequence; or f] co{xk: k J? n} ^ 0). And we say that (X, || ||) has the ANP-K for K =1, II or III if there is a norming set $ for (X, || ||) such that X has the S-ANP-K. The space X has the ANP-K for K = I, II or III if there is an equivalent norm || || on X such that (X, \\ ||) has the ANP-K.

“…(This property in clearly weaker than the Mazur intersection property and is also weaker than the Gateaux-smoothness of the norm [3] and [5].) Moreover we give a more precise result: we prove that the convex compact subset of H which fails to be an intersection of balls can be choosen to be of affiiie dimension at most equal to 2.…”

Section: Introductionmentioning

confidence: 81%

Smoothness in spaces of compact operators

Sersouri

1988

SMOOTHNESS IN SPACES OF COMPACT OPERATORSA. SERSOURIWe prove that if X and Y are two (real) Banach spaces such that dim X ^ 2 and dim Y ^ 2, then the space K(X, Y) contains a convex compact subset C with dim G ^ 2 (in the affiiie sense) which fails to be an intersection of balls. This improves two results of Ruess and Stegall. INTRODUCTIONIn this paper we consider only real Banach spaces. This is not a restriction since all the properties considered in this paper depend only on the real structure of the space.It was proved by Ruess and Stegall [2] that if H is a subspace of K w *(X*,Y) (that is, the space of compact IU* -w continuous operators) winch contains X ® Y, then whenever dimX ^ 2 and dimY" Js 2, the space H never has the Mazur intersection property (that is, every bounded closed convex set is the intersection of the (closed) balls containing it), and is never Gateaux-smooth (see also [1] Theorem 2.1).In this paper we improve the results of Ruess and Stegall by proving that under the same assumptions as above, the space H never has the property (CI): every convex compact set is the intersection of the balls containing it. (This property in clearly weaker than the Mazur intersection property and is also weaker than the Gateaux-smoothness of the norm [3] and [5].) Moreover we give a more precise result: we prove that the convex compact subset of H which fails to be an intersection of balls can be choosen to be of affiiie dimension at most equal to 2.This result cannot -in general -be improved, since every line segment of the space t\ <8> t £\ ' s a n intersection of balls. NOTATIONFor a finite dimensional convex set C we denote by dimC the affiiie dimension of C.A point a: of a Banach space X is said to be an extreme point if x = 0 or a;/1| ar || is an extreme point of the unit ball B(X) of X. The set of extreme points of X will be denoted by Ext(X), By a ball we always mean a closed ball. RESULTS