Unconditional families in Banach spaces

Argyros, Spiros A.; Dodos, Pandelis; Kanellopoulos, Vassilis

doi:10.1007/s00208-007-0179-y

Cited by 32 publications

(41 citation statements)

References 41 publications

(61 reference statements)

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“…Hence by (3) M c / ∈ G and so necessarily M ∈ G. This proves (2). We now prove (5) by induction on k. For k = 1 property (5) is trivial.…”

Section: Lemma 12 G Decides Amentioning

confidence: 73%

“…In their recent study on Banach spaces not containing 1 in [2] the authors were led to consider compact spaces of Borel functions which are not necessarily separable but still admit Σ 1 1 codings relatively to some sequences. We now state in this more general frame some consequences of (A * ).…”

Section: Remarks 19 (A)mentioning

confidence: 99%

“…The problems discussed in the present work concern the descriptive properties of these spaces. This was actually initiated in [4], but it is only quite recently that the interest of such a study was highlighted first by the work of Dodos in [5] and since then by several other interesting applications (see [1,2,7]). In fact our initial motivation was to solve a natural problem left open in our earlier study [4].…”

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confidence: 98%

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Borel extractions of converging sequences in compact sets of Borel functions

Debs

2009

Journal of Mathematical Analysis and Applications

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It is well known by a classical result of Bourgain-Fremlin-Talagrand that if K is a pointwise compact set of Borel functions on a Polish space then given any cluster point f of a sequence (f n ) n∈ω in K one can extract a subsequence (f n k ) k∈ω converging to f . In the present work we prove that this extraction can be achieved in a "Borel way." This will prove in particular that the notion of analytic subspace of a separable Rosenthal compacta is absolute and does not depend on the particular choice of a dense sequence. This paper is written in a double aim: firstly to prove the results announced in the abstract above, secondly to emphasize the interest of "effective" methods in the treatment of classical descriptive problems. The paper is indeed conceived as an invitation to the non specialist reader for the use of these powerful methods. And though a thorough reading of the paper necessitates clearly a great familiarity with Effective Descriptive Set Theory (for example as developed in [9]) our deep hope is that the text will be comprehensible, at least in its general framework, by a reader with a standard classical background. To that end we shall give in Section 1 a quick informal presentation of the main effective concepts and results used in this work.The basic topological properties of Rosenthal compacta, or more generally of compact spaces of Borel functions on a given Polish space, were entirely elucidated by the fundamental works of Rosenthal [10] and later on Bourgain, Fremlin and Talagrand [3]. The problems discussed in the present work concern the descriptive properties of these spaces. This was actually initiated in [4], but it is only quite recently that the interest of such a study was highlighted first by the work of Dodos in [5] and since then by several other interesting applications (see [1,2,7]). In fact our initial motivation was to solve a natural problem left open in our earlier study [4]. Incidently the solution of this old problem

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“…Hence by (3) M c / ∈ G and so necessarily M ∈ G. This proves (2). We now prove (5) by induction on k. For k = 1 property (5) is trivial.…”

Section: Lemma 12 G Decides Amentioning

confidence: 73%

Section: Remarks 19 (A)mentioning

confidence: 99%

mentioning

confidence: 98%

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Borel extractions of converging sequences in compact sets of Borel functions

Debs

2009

Journal of Mathematical Analysis and Applications

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“…For the equivalence (2) and (3) below (and applications) in the class of locally convex spaces we refer the reader to [41], [62] and [63]. The equivalence between (1) and (3) is due to Saxon-Wilansky [65]. Recall that a locally convex space E is barrelled if every barrel (an absolutely convex closed and absorbing set in E) is a neighborhood of zero.…”

Section: A Few Results For General Banach Spacesmentioning

confidence: 99%

On the separable quotient problem for Banach spaces

Ferrando¹,

Ka̧kol²,

Pellicer³

et al. 2018

Funct. Approx. Comment. Math.

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While the classic separable quotient problem remains open, we survey general results related to this problem and examine the existence of a particular infinitedimensional separable quotient in some Banach spaces of vector-valued functions, linear operators and vector measures. Most of the results presented are consequence of known facts, some of them relative to the presence of complemented copies of the classic sequence spaces c 0 and ℓ p , for 1 ≤ p ≤ ∞. Also recent results of Argyros, Dodos, Kanellopoulos [1] andŚliwa [66] are provided. This makes our presentation supplementary to a previous survey (1997) due to Mujica. 1991 Mathematics Subject Classification. 46B28, 46E27, 46E30.

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“…The above proposition is based on Theorem 6 and on the following recent structural result ( [3], see also [2]) which is a key ingredient for our proof.…”

Section: Propositionmentioning

confidence: 99%

On the subspaces of JF and JT with non-separable dual

Apatsidis

Argyros

Kanellopoulos

2008

Journal of Functional Analysis

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The connectivity of the hydrophobic medium in the nonionic binary system C 12 EO 6 /H 2 O is studied by monitoring the diffusion constants of tracer molecules at the transition between the hexagonal mesophase and the fluid isotropic phase. The increase in the transverse diffusion coefficient on approaching the isotropic phase reveals the proliferation of bridge-like defects connecting the surfactant cylinders. This suggests that the isotropic phase has a highly connected structure. Indeed, we find similar diffusion coefficients in the isotropic and cubic bicontinuous phases. The temperature dependence of the lattice parameter in the hexagonal phase confirms the change in connectivity close to the hexagonal-isotropic transition. Finally, an X-ray investigation of the isotropic phase shows that its structure is locally similar to that of the hexagonal phase.

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Unconditional families in Banach spaces

Cited by 32 publications

References 41 publications

Borel extractions of converging sequences in compact sets of Borel functions

Borel extractions of converging sequences in compact sets of Borel functions

On the separable quotient problem for Banach spaces

On the subspaces of JF and JT with non-separable dual

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