Weakly extremal properties of Banach spaces (original) (raw)

Extension of the Erberlein-Smulian Theorem to Normed Spaces

2005

The Erberlein-Smulian theorem asserts that for complete normed spaces, that is, Banach spaces, a subset is weak compact if and only if it is weak sequentially compact. In this paper it is shown that the completeness of the normed space is not necessary for the above mentioned result. Here we establish the weak compactness of a bounded subset in a normed space, not necessarily complete, by using Alaoglu's theorem: Alaoglu's theorem. For any normed linear space X, B X * , the closed unit ball in the dual X * , is weak * compact. Consequently, weak * closed and (norm) bounded subsets of the normed space X * are weak * compact. Let A be a bounded subset of a normed space X. To showĀ weak is weak compact, that is, A is relatively weak compact, we need to see howĀ weak looks like. Consider A as J[A] where J is the canonical embedding of X into X * *. Let us look atĀ weak * in the bidual X * * .Ā weak is identical toĀ weak * , provided that the elements in X * * found inĀ weak * are precisely point evaluational functionals, that is, in X. Furthermore, the weak * and the weak topologies coincide. Hence, once we show thatĀ weak * is weak * compact by using Alaoglu's theorem, then A weak will also be weak compact. Proposition 1.Ā weak * is weak * compact. Proof. By Alaoglu's theorem, to showĀ weak * is weak * compact it suffices to show thatĀ weak * is (norm) bounded in the normed space X * *. By Lemma 1 (please see the appendix), applied to the normed linear space X * * , it is equivalent to show thatĀ weak * is weakly bounded (as defined in Lemma 1) in the bidual X * *. We therefore show that: (∀f ∈ X * * *) (sup x * * ∈Ā weak * {| f (x * *) | ≤ K f }) where K f is a constant which depends on f .

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.

A note on extreme points in dual spaces

Acta Mathematica Sinica, English Series, 2013

Given a normed space X it can be easily proven that every extreme point in B X * , the unit ball of X * , is the restriction of an extreme point in B X * * *. Our purpose is to study when the restrictions of extreme points in B X * * * are extreme points in B X *. Namely, we characterize L 1-preduals satisfying the aforementioned property.

On weak*-extensible Banach spaces

Nonlinear Analysis: Theory, Methods & Applications, 2012

We study the stability properties of the class of weak*-extensible spaces introduced by Wang, Zhao, and Qiang showing, among other things, that weak*-extensibility is equivalent to having a weak*-sequentially continuous dual ball (in short, w*SC) for duals of separable spaces or twisted sums of w*SC spaces. This shows that weak*-extensibility is not a 3-space property, solving a question posed by Wang, Zhao, and Qiang. We also introduce a restricted form of weak*-extensibility, called separable weak*-extensibility, and show that separably weak*-extensible Banach spaces have the Gelfand-Phillips property, although they are not necessarily w*SC 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.

Weakly open sets in the unit ball of some Banach spaces and the centralizer

Journal of Functional Analysis, 2010

We show that every Banach space X whose centralizer is infinite-dimensional satisfies that every nonempty weakly open set in B Y has diameter 2, where Y = N,s,π X (N-fold symmetric projective tensor product of X, endowed with the symmetric projective norm), for every natural number N. We provide examples where the above conclusion holds that includes some spaces of operators and infinite-dimensional C *-algebras. We also prove that every non-empty weak * open set in the unit ball of the space of Nhomogeneous and integral polynomials on X has diameter two, for every natural number N , whenever the Cunningham algebra of X is infinite-dimensional. Here we consider the space of N-homogeneous integral polynomials as the dual of the space N,s,ε X (N-fold symmetric injective tensor product of X, endowed with the symmetric injective norm). For instance, every infinite-dimensional L 1 (μ) satisfies that its Cunningham algebra is infinite-dimensional. We obtain the same result for every non-reflexive L-embedded space, and so for every predual of an infinite-dimensional von Neumann algebra.

On the drop and weak drop properties for a Banach space

Bulletin of the Australian Mathematical Society, 1990

Rolewicz' drop property is a modification of a concept underlying Danes' drop theorem. We characterise the drop property by the upper semicontinuity and compact valued property of the duality mapping for the dual. The characterisation suggests that we define a weak drop property which we show characterises the reflexivity of the space. Consider the Banach space (X, ||-||) with open unit ball B(X) = {x 6 X: \\x\\ < 1} and closed unit ball B(X) = {x G X: \\x\\ < 1}. Given x £ B~(X), the set D(x, ~B(X)) = co{x, ~B(X)}, the convex hull of x and ~B(X), is called the drop generated by x. Danes [3] proved that in any Banach space (X, \\-\\), for every closed set C at positive distance from B(X), there exists an x G C such that D(x, -B(Z)) fl C = {x}. Rolewicz [9], modifying the Danes' drop theorem assumption, said that the norm ||-|| of X has the drop property if for every closed set C disjoint from B~{X), there exists an x G C such that D(x, ~B(XJ) ClC = {x}. He also introduced an associated sequential concept; a sequence {x n } in Jf \ B(X) such that x n +i £ D(x n , B(X)) for all n, is called a stream. Rolewicz proved that (i) the norm ||-|| has the drop property if and only if each stream in X\B(X) contains a convergent subsequence, [9, Proposition 2, p.29), and (ii) if the norm ||-|| has the drop property then X is reflexive, [9, Theorem 5, p.34].

3 on the Bounded Approximation Property in Banach Spaces

2016

We prove that the kernel of a quotient operator from an L 1-space onto a Banach space X with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky-case ℓ 1-and Figiel, Johnson and Pe lczyński-case X * separable. Given a Banach space X, we show that if the kernel of a quotient map from some L 1-space onto X has the BAP then every kernel of every quotient map from any L 1-space onto X has the BAP. The dual result for L∞-spaces also hold: if for some L∞-space E some quotient E/X has the BAP then for every L∞-space E every quotient E/X has the BAP.