A Biased View of Topology as a Tool in Functional Analysis

Cascales, B.; Orihuela, J.

doi:10.2991/978-94-6239-024-9_3

Cited by 9 publications

(10 citation statements)

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“…[30] (see also [2]). The concept of network is one of a well recognized good tool, coming from the pure set-topology, which turned out to be of great importance to study successfully renorming theory in Banach spaces, we refer the reader to the recent survey of Cascales and Orihuela [12].…”

Section: Introductionmentioning

confidence: 99%

On topological properties of Fréchet locally convex spaces with the weak topology

Gabriyelyan

Ka̧kol

Kubzdela

et al. 2015

Topology and its Applications

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Elsevier Gabriyelyan, S.; Kakol, JM.; Kubzdela, A.; López Pellicer, M. (2015). On topological properties of Fréchet locally convex spaces. Topology and its Applications. 192 AbstractWe describe the topology of any cosmic space and any ℵ 0 -space in terms of special bases defined by partially ordered sets. Using this description we show that a Baire cosmic group is metrizable. Next, we study those locally convex spaces (lcs) E which under the weak topology σ(E, E ) are ℵ 0 -spaces. For a metrizable and complete lcs E not containing (an isomorphic copy of) 1 and satisfying the Heinrich density condition we prove that (E, σ(E, E )) is an ℵ 0 -space if and only if the strong dual of E is separable. In particular, if a Banach space E does not contain 1 , then (E, σ(E, E )) is an ℵ 0 -space if and only if E is separable. The last part of the paper studies the question: Which spaces (E, σ(E, E )) are ℵ 0 -spaces? We extend, among the others, Michael's results by showing: If E is a metrizable lcs or a (DF )-space whose strong dual E is separable, then (E, σ(E, E )) is an ℵ 0 -space. Supplementing an old result of Corson we show that, for ǎ Cech-complete Lindelöf space X the following are equivalent: (a) X is Polish, (b) C c (X) is cosmic in the weak topology, (c) the weak * -dual of C c (X) is an ℵ 0 -space.

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Section: Introductionmentioning

confidence: 99%

On topological properties of Fréchet locally convex spaces with the weak topology

Gabriyelyan

Ka̧kol

Kubzdela

et al. 2015

Topology and its Applications

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“…33 Surveys about these questions are. 24,40,41 3. Upper semi-continuity for multifunctions, applications…”

Section: Fig 4 a Sandwich Resultsmentioning

confidence: 99%

“…Proof. We refer to the proof of this theorem to [40,Theorem 2.6]. We reproduce here only the implication (7) ⇒ (2).…”

Section: Corollary 31 ( [43 Corollary 11])mentioning

confidence: 99%

Measurability and semi-continuity of multifunctions

Cascales

2016

Advanced Courses of Mathematical Analysis V

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“…On the other hand, the existence of an equivalent σ(X, F )-LUR norm was characterized by the second named author in terms of countable decompositions of the space X, as well as network conditions close to covering properties of generalized metrizable spaces, (see [1] and [7]). Moreover, as we shall see here, it is somehow natural to think that the existence of a σ(X, F )-LUR renorming could be characterized by covering properties of the sphere closed to its metrizability for the σ(X, F )-topology.…”

Section: Introductionmentioning

confidence: 99%