Limited Sets in <i>C</i>(<i>K</i>)‐Spaces and Examples Concerning the Gelfand‐Phillips‐Property

Schlumprecht, Thomas

doi:10.1002/mana.19921570105

Cited by 13 publications

(17 citation statements)

References 8 publications

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“…D is in the class if (and only if) there is, for every b 0, a set D in the class such that D & D B E . In [8] this analogy of Grothendieck's characterization of the relatively weakly compact sets is observed to hold for limited sets. It is easily verified that the classes of compact, limited and uniformly limited sets have both properties.…”

Section: Introductionmentioning

confidence: 76%

“…Remark 2. In [4] and [8] it is observed that E is a GP-space if the dual ball B 1 contains a sequentially weak c -precompact E-norming subset B, i.e., there exists c b 0 so that kzk c sup 9PB j9zj for every z P E. A direct proof of this gives that if D fe n g & E satisfies ke n k ! d and d ke n À e j k for a given d b 0 and all n T j, then there exists a weak c null sequence 9 n & 8cB 1 with lim sup j9 n e n j !…”

Section: Uniform Boundsmentioning

confidence: 99%

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Uniform Bounds for Limited Sets and Applications to Bounding Sets.

Josefson¹

2000

MATH. SCAND.

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A set D in a Banach space E is limited if lim sup k3I sup zPD j9 k zj b 0 Asup kzk1 lim sup k3I j9 k zjb 0 for every sequence 9 k & E c. It is studied how this implication can be quantified, for example if there exists a constant C b 0 such that lim sup k3I sup zPD j9 k zj 1 A sup kzk1 lim sup k3I j9 k zj ! C for every sequence 9 k & E c , is studied. Relatively compact sets and limited sets in l I-among others the unit vectors-have uniform bounds in this sense. A fundamental example of a limited set without any uniform bounds is constructed. A set D is called bounding if f D is bounded for every entire function on E. That bounding sets are uniformly limited and that strongly bounding sets are limited in the strongest sense are proved. Examples show that the convex hull of bounding sets in general are not bounding and that bounding sets in general does not have Grothendieck's incapsulating property as relatively weakly compact sets have. lim k3I sup zPD j9 k zj 0 An equivalent formulation without referring to null sequences is: D is called limited if lim sup k3I sup zPD j9 k zj b 0 A sup kzk1 lim sup k3I j9 k zj b 0 for every sequence 9 k & E c. A natural question is how this relation can be quantified. For example we may ask if a limited set D has a constant C b 0 such that lim sup k3I sup zPD j9 k zj 1 A sup kzk1 lim sup k3I j9 k zj ! C for every sequence 9 k & E c. Such a limited set is called limited in the

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Section: Introductionmentioning

confidence: 76%

Section: Uniform Boundsmentioning

confidence: 99%

Uniform Bounds for Limited Sets and Applications to Bounding Sets.

Josefson¹

2000

MATH. SCAND.

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“…Building on a result due to Schlumprecht [28], we give in the final section an affirmative solution to Problem 1.2. Our construction, however, relies on a set-theoretic assumption, whose consistency has not yet been established.…”

Section: Introductionmentioning

confidence: 81%

“…The Gelfand-Phillips property has attracted considerable attention over the last twenty years, which resulted in several interesting papers, see for instance Bourgain & Diestel [5], Drewnowski [6], Schlumprecht [28], Sinha & Arora [26], Freedman [9]. The class (GP) of spaces having this property is quite wide, and includes (i) l 1 (κ) for every κ;…”

Section: Mazur Versus Gelfand-phillipsmentioning

confidence: 99%

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On sequential properties of Banach spaces, spaces of measures and densities

Borodulin–Nadzieja

Plebanek

2010

Czech Math J

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We answer in negative the problem if the existence of a Pmeasure implies the existence of a P-point. Namely, we show that if we add random reals to a certain 'unique P-point' model, then in the resulting model we will have a P-measure but not P-points. Also, we investigate the question if there is a P-measure in the Silver model. We show that rapid filters cannot be extended to a P-measure in the extension by ω product of Silver forcings and that in the model obtained by the countable support ω 2 -iteration of countable product of Silver forcings there are no P-measures of countable Maharam type.

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The b-Gelfand–Phillips property for locally convex spaces

Банах

Gabriyelyan

2023

Collect. Math.

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Limited Sets in C(K)‐Spaces and Examples Concerning the Gelfand‐Phillips‐Property

Cited by 13 publications

References 8 publications

Uniform Bounds for Limited Sets and Applications to Bounding Sets.

Uniform Bounds for Limited Sets and Applications to Bounding Sets.

On sequential properties of Banach spaces, spaces of measures and densities

The b-Gelfand–Phillips property for locally convex spaces

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