Locally convex spaces not containing $l^1$

Ruess, Wolfgang M.

doi:10.7169/facm/2014.50.2.9

Cited by 10 publications

(8 citation statements)

References 15 publications

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“…As shown in Theorem 7.6, "Tame dichotomy" of Theorem 7.2 implies that Rosenthal's classical dichotomy from Banach spaces can be extended to a much larger class of lcs with metrizable bounded subsets. This strengthens a well-known result of Ruess [47] (Fact 7.8). On the other hand, Rosenthal's dichotomy is not true for general lcs as Theorem 7.9 demonstrates.…”

Section: Rosenthal Type Propertiessupporting

confidence: 90%

Tameness and Rosenthal type locally convex spaces

Matan¹,

Megrelishvili²

2022

Preprint

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Motivated by Rosenthal's famous l 1 -dichotomy in Banach spaces, Haydon's theorem, and additionally by recent works on tame dynamical systems, we introduce the class of tame locally convex spaces. This is a natural locally convex analog of Rosenthal Banach spaces (for which any bounded sequence contains a weak Cauchy subsequence). Our approach is based on a bornology of tame subsets which in turn is closely related to eventual fragmentability. This leads, among others, to the following results:• extending Haydon's characterization of Rosenthal Banach spaces, by showing that a lcs E is tame iff every weak-star compact, equicontinuous convex subset of E * is the strong closed convex hull of its extreme points iff co w * (K) = co (K) for every weak-star compact equicontinuous subset K of E * ; • E is tame iff there is no bounded sequence equivalent to the generalized l 1 -sequence;• strengthening some results of W.M. Ruess about Rosenthal's dichotomy;• applying the Davis-Figiel-Johnson-Pelczyński (DFJP) technique one may show that every tame operator T : E → F between lcs can be factored through a tame lcs.

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Section: Rosenthal Type Propertiessupporting

confidence: 90%

Tameness and Rosenthal type locally convex spaces

Matan¹,

Megrelishvili²

2022

Preprint

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“…However, in general the Schur property does not imply the weak respecting compactness (see [42,Example 6 (p. 267)] and [14,Example 19.19] or the more general Proposition 3.5 below), and hence the condition of being metrizable for separable bounded sets in Theorem 1.2 is essential. We prove Theorem 1.2 in Section 2 using (1) an extension of the Rosenthal ℓ 1 theorem obtained recently by W. Ruess [39], and (2) the aforementioned result of M. Valdivia [40]. As a corollary of Theorem 1.2 we provide numerous characterizations of the Schur property for strict (LF )-spaces, see Corollary 2.13.…”

Section: Introductionmentioning

confidence: 87%

“…In [11, Lemma 3] J.C. Díaz extends the Rosenthal ℓ 1 theorem to all Fréchet spaces. A much more general result was obtained recently by W. Ruess in [39]. Recall that an lcs E is locally complete if every closed disc in E is a Banach disc; every sequentially complete lcs is locally complete by Corollary 5.1.8 of [35].…”

Section: Proof Of Theorem 12mentioning

confidence: 97%

Locally convex spaces and Schur type properties

Gabriyelyan¹

2019

Ann. Acad. Sci. Fenn. Math.

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In the main result of the paper we extend Rosenthal's characterization of Banach spaces with the Schur property by showing that for a quasi-complete locally convex space E whose separable bounded sets are metrizable the following conditions are equivalent: (1) E has the Schur property, (2) E and Ew have the same sequentially compact sets, where Ew is the space E with the weak topology, (3) E and Ew have the same compact sets, (4) E and Ew have the same countably compact sets, (5) E and Ew have the same pseudocompact sets, (6) E and Ew have the same functionally bounded sets, (7) every bounded non-precompact sequence in E has a subsequence which is equivalent to the unit basis of ℓ1 and (8) every bounded non-precompact sequence in E has a subsequence which is discrete and C-embedded in Ew.2000 Mathematics Subject Classification. 46A03, 46E10.

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“…Let E be a metrizable lcs. Then every bounded subset of E is Fréchet-Urysohn in the weak topology of E if and only if every bounded sequence in E has a Cauchy subsequence in the weak topology of E. Proposition 6.2 ( [31]). Let E be a complete lcs such that every bounded set in E is metrizable.…”

Section: Proof Of Theorem 110 and Final Questionsmentioning

confidence: 99%

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

2016

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Following [3] we say that a Tychonoff space X is an Ascoli space if every compact subset K of C k (X) is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every k R -space, hence any k-space, is Ascoli.Let X be a metrizable space. We prove that the space C k (X) is Ascoli iff C k (X) is a k R -space iff X is locally compact. Moreover, C k (X) endowed with the weak topology is Ascoli iff X is countable and discrete.Using some basic concepts from probability theory and measure-theoretic properties of ℓ 1 , we show that the following assertions are equivalent for a Banach space E: (i) E does not contain isomorphic copy of ℓ 1 , (ii) every real-valued sequentially continuous map on the unit ball B w with the weak topology is continuous,We prove also that a Fréchet lcs F does not contain isomorphic copy of ℓ 1 iff each closed and convex bounded subset of F is Ascoli in the weak topology. However we show that a Banach space E in the weak topology is Ascoli iff E is finite-dimensional. We supplement the last result by showing that a Fréchet lcs F which is a quojection is Ascoli in the weak topology iff either F is finite dimensional or F is isomorphic to the product K N , where K ∈ {R, C}.

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Locally convex spaces not containing $l^1$

Cited by 10 publications

References 15 publications

Tameness and Rosenthal type locally convex spaces

Tameness and Rosenthal type locally convex spaces

Locally convex spaces and Schur type properties

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

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