Envelopes of open sets and extending holomorphic functions on dual Banach spaces

Garcı́a, Domingo; Kalenda, Ondřej F. K.; Maestre, Manuel

doi:10.1016/j.jmaa.2009.09.051

Cited by 4 publications

(4 citation statements)

References 18 publications

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“…Proof. We know that vectors (11) are uniformly bounded. Because of this it is enough to prove that the vectors (11) converge to the vector (12) componentwise.…”

Section: Proof Of Item (B)mentioning

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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“…Proof. We know that vectors (11) are uniformly bounded. Because of this it is enough to prove that the vectors (11) converge to the vector (12) componentwise.…”

Section: Proof Of Item (B)mentioning

confidence: 99%

“…Garcia-Kalenda-Maestre [11] initiated the theory of weak * derived sets for convex sets. This theory was developed by Ostrovskii [29], who proved that, for an arbitrary nonreflexive Banach space X (not necessarily non-quasi-reflexive), there a convex set A ⊂ X for which A (1) = A (2) .…”

Section: Introductionmentioning

confidence: 99%

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

Ostrovskii¹

2021

Preprint

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“…Further, the orders of subspaces in a dual to a separable Banach space must be countable and cannot be limit; see for example [6]. Later, García, Kalenda and Maestre [5] asked the following questions in their paper about extension problems for holomorphic functions on dual Banach spaces.…”

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confidence: 99%

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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“…The fact that the existing theory of Krein-Šmulian together with the examples listed above for subspaces does not contain answers to all questions which are natural to ask about convex sets was noticed by García-Kalenda-Maestre [11] in their study of extension problems for holomorphic functions on dual Banach spaces. They initiated a further development of the theory for convex sets by asking the following question [11,Question 6.3]: Let X be a quasi-reflexive Banach space. Is A (1) equal to A * for each (absolutely) convex set A ⊂ X ?…”

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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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Envelopes of open sets and extending holomorphic functions on dual Banach spaces

Cited by 4 publications

References 18 publications

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

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

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

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

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