Weak⁎ derived sets of convex sets in duals of non-reflexive spaces

Silber, Zdeněk

doi:10.1016/j.jfa.2021.109259

Cited by 3 publications

(7 citation statements)

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“…for every convex subset A of X * if and only if X is reflexive. This result was later developed by the author [16] and Ostrovskii [8] in the spirit of Theorem 1.1. Let us note that it is still an open problem whether the order of a convex set can be a countable limit ordinal.…”

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confidence: 87%

On subspaces whose weak$^*$ derived sets are proper and norm dense

Silber

2023

Studia Math.

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We study long chains of iterated weak * derived sets, that is, sets of all weak * limits of bounded nets, of subspaces with the additional property that the penultimate weak * derived set is a proper norm dense subspace of the dual. We extend the result of Ostrovskii and show that in the dual of any non-quasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual, we can find for any countable successor ordinal α a subspace whose weak * derived set of order α is proper and norm dense.

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confidence: 87%

On subspaces whose weak$^*$ derived sets are proper and norm dense

Silber

2023

Studia Math.

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“…Our construction of sets is quite different from the one in [29,42]. For our construction of sets whose existence is stated in Theorem 1.1, we need families of both finite and infinite forests -that is, graphs with no cycles, and trees -that is, connected forests.…”

Section: Trees and Forestsmentioning

confidence: 99%

“…We have the convergence of To complete the proof of item (B), it suffices: of infinitely many disjoint copies of F 1 . We do not provide the details because the case α = 1 is covered by Silber [42].…”

Section: Proof Of Item (B)mentioning

confidence: 99%

“…The following question asked in [42] remains unanswered. Question 5.7 ([42, Section 3, Question 1]).…”

Section: Commentmentioning

confidence: 99%

“…Silber [42] developed this result further by proving that, for an arbitrary non-reflexive space X and an arbitrary n ∈ N, the space X * contains a convex subset A for which n is the least ordinal satisfying A (n) = A * and a convex subset D for which ω + 1 is the least ordinal satisfying D (α) = D * . Our goal is to develop this theory further by proving the following result.…”

Section: Introductionmentioning

confidence: 99%

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Weak$^*$ closures and derived sets for convex sets in dual Banach spaces

Ostrovskii¹

2021

Preprint

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Weak$^*$ closures and derived sets for convex sets in dual Banach spaces

Ostrovskii

2023

Studia Math.

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The paper is devoted to the convex-set counterpart of the theory of weak * derived sets initiated by Banach and Mazurkiewicz for subspaces. The main result is the following: For every nonreflexive Banach space X and every countable successor ordinal α, there exists a convex subset A in X * such that α is the least ordinal for which the weak * derived set of order α coincides with the weak * closure of A. This result extends the previously known results on weak * derived sets by Ostrovskii (2011) and Silber (2021).1. Introduction. Let X be a Banach space. For a subset A of the dual Banach space X * , we denote the weak * closure of A by A * . The weak * derived set of A is defined aswhere B X * is the unit ball of X * . That is, A (1) is the set of all limits of weak * convergent bounded nets in A. If X is separable, A (1) coincides with the set of all limits of weak * convergent sequences from A, called the weak * sequential closure. The strong closure of a set A in a Banach space is denoted by A. We set A (0) := A.It was noticed in the early days of Banach space theory by Mazurkiewicz [23] that A (1) does not have to coincide with A * even for a subspace A, and (A (1) ) (1) can be different from A (1) . In this connection, it is natural to introduce derived sets for all ordinals: (1) if A (α) has already been defined, then A (α+1) := (A (α) ) (1) ; (2) if α is a limit ordinal and A (β) has already been defined for all β < α, then(1.1) β) .

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Weak⁎ derived sets of convex sets in duals of non-reflexive spaces

Cited by 3 publications

References 8 publications

On subspaces whose weak$^*$ derived sets are proper and norm dense

On subspaces whose weak$^*$ derived sets are proper and norm dense

Weak$^*$ closures and derived sets for convex sets in dual Banach spaces

Weak$^*$ closures and derived sets for convex sets in dual Banach spaces

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