Non-dentable sets in Banach spaces with separable dual (original) (raw)

Cantor sets in the dual of a separable Banach space. Applications

This survey collects several (classical and more recent) results relating some structural properties of a Banach space X (Asplundness, containing a copy of 1,...) with the existence of a subset of the unit ball BX * of the dual that, in the weak star topology, is homeomorphic to the ternary Cantor set. The possibility of finding this set inside the extreme points of B X * is also considered. Some applications are described.

On Hereditarily Indecomposable Banach Spaces

Acta Mathematica Sinica, English Series, 2006

This paper shows that every non-separable hereditarily indecomposable Banach space admits an equivalent strictly convex norm, but its bi-dual can never have such a one; consequently, every non-separable hereditarily indecomposable Banach space has no equivalent locally uniformly convex norm.

Reflexivity and the Separable Quotient Problem for a Class of Banach Spaces

Bulletin of the Polish Academy of Sciences Mathematics

Let E be a Banach lattice and let X be its closed subspace such that: X is complemented in E, or the norm of E is order continuous. Then X is reflexive iff X* contains no isomorphic copy of \ell_1 iff for every n ≥ 1, the nth dual X^(n) of X contains no isomorphic copy of \ell_1 iff X has no quotient isomorphic to c_0 and X does not have a subspace isomorphic to \ell_1 (Theorem 2). This is an extension of the results obtained earlier by Lozanovski˘ i, Tzafriri, Meyer-Nieberg, and Diaz and Fern´andez. The theorem is applied to show that many Banach spaces possess separable quotients isomorphic to one of the following spaces: c_0, \ell_1, or a reflexive space with a Schauder basis.

Nondentable Sets in Banach Spaces

2020

In his study of the Radon Nikod\'ym property of Banach spaces, Bourgain showed (among other things) that in any closed, bounded, convex set AAA that is nondentable, one can find a separated, weakly closed bush. In this note, we prove a generalization of Bourgain's result: in any bounded, nondentable set AAA (not necessarily closed or convex) one can find a separated, weakly closed approximate bush. Similarly, we obtain as corollaries the existence of AAA-valued quasimartingales with sharply divergent behavior.

Some strongly bounded classes of Banach spaces

Fundamenta Mathematicae, 2007

We show that the classes of separable reflexive Banach spaces and of spaces with separable dual are strongly bounded. This gives a new proof of a recent result of E. Odell and Th. Schlumprecht, asserting that there exists a separable reflexive Banach space containing isomorphic copies of every separable uniformly convex Banach spaces.

On separably injective Banach spaces

Advances in Mathematics, 2013

In this paper we deal with two weaker forms of injectivity which turn out to have a rich structure behind: separable injectivity and universal separable injectivity. We show several structural and stability properties of these classes of Banach spaces. We provide natural examples of (universally) separably injective spaces, including L∞ ultraproducts built over countably incomplete ultrafilters, in spite of the fact that these ultraproducts are never injective. We obtain two fundamental characterizations of universally separably injective spaces: a) A Banach space E is universally separably injective if and only if every separable subspace is contained in a copy of ℓ∞ inside E. b) A Banach space E is universally separably injective if and only if for every separable space S one has Ext(ℓ∞/S, E) = 0. The final Section of the paper focuses on special properties of 1-separably injective spaces. Lindenstrauss obtained in the middle sixties a result that can be understood as a proof that, under the continuum hypothesis, 1-separably injective spaces are 1-universally separably injective; he left open the question in ZFC. We construct a consistent example of a Banach space of type C(K) which is 1-separably injective but not 1-universally separably injective.

On rrr-reflexive Banach spaces

2009

A Banach space XXX is called {\it rrr-reflexive\/} if for any cover CalU\Cal UCalU of XXX by weakly open sets there is a finite subfamily CalVsubsetCalU\Cal V\subset\Cal UCalVsubsetCalU covering some ball of radius 1 centered at a point xxx with ∣x∣leqr\|x\|\leq r∣x∣leqr. We prove that an infinite-dimensional separable Banach space XXX is infty\inftyinfty-reflexive ($r$-reflexive for some rinBbbNr\in \Bbb NrinBbbN) if and only if each varepsilon\varepsilon varepsilon-net for XXX has an accumulation point (resp., contains a non-trivial convergent sequence) in the weak topology of XXX. We show that the quasireflexive James space JJJ is rrr-reflexive for no rinBbbNr\in \Bbb NrinBbbN. We do not know if each infty\inftyinfty-reflexive Banach space is reflexive, but we prove that each separable infty\inftyinfty-reflexive Banach space XXX has Asplund dual. As a by-product of the proof we obtain a covering characterization of the Asplund property of Banach spaces.

A Characterization of Banach Spaces with Separable Duals via Weak Statistical Convergence

Journal of Mathematical Analysis and Applications, 2000

Let B be a Banach space. A B-valued sequence x k is weakly statistically null provided lim n 1 n k ≤ n f x k > ε = 0 for all ε > 0 and every continuous linear functional f on B. A Banach space is finite dimensional if and only if every weakly statistically null B-valued sequence has a bounded subsequence. If B is separable, B * is separable if and only if every bounded weakly statistically null B-valued sequence contains a large weakly null sequence. A characterization of spaces containing an isomorphic copy of l 1 is given, and it is also shown that l 2 has a "statistical M-basis" which is not a Schauder basis.

A Banach space not containing l1l_{1}l1 whose dual ball is not weak${}^ ast$ sequentially compact

Illinois Journal of Mathematics, 1978

balls was studied in , where it was proved that if X is separable and the unit ball of X** is not weak* sequentially compact, then X* contains a subspace isomorphic to 11 (F) for some uncountable set F. Subsequently it was proved in [1] that if the unit ball of X* is not weak* sequentially compact, then (a)either Co is a quotient of X or 11 is isomorphic to a subspace of X, and (b) X has a separable subspace with nonseparable dual. In this note we give an example of a Banach space X whose dual ball is not weak* sequentially compact, but where X contains no subspace isomorphic to t. This answers a question posed by H. P. Rosenthal .

On Banach spaces whose dual balls are not weak∗ sequentially compact

Israel Journal of Mathematics, 1977

THEOREM 1. Let X be a Banach space. (a) If X* has a closed subspace in which no normalized sequence converges weak* to zero, then I, is isomorphic to a subspace of X. (b) If X* contains a bounded sequence which has no weak* convergent subsequence, then X contains a separable subspace whose dual is not separable.