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1
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Remark on the Abstract Dirichlet Problem for Baire-One Functions

100%
Kalenda O.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2005
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tom 53
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nr 1
55-73
EN
We study the possibility of extending any bounded Baire-one function on the set of extreme points of a compact convex set to an affine Baire-one function and related questions. We give complete solutions to these questions within a class of Choquet simplices introduced by P. J. Stacey (1979). In particular we get an example of a Choquet simplex such that its set of extreme points is not Borel but any bounded Baire-one function on the set of extreme points can be extended to an affine Baire-one function. We also study the analogous questions for functions of higher Baire classes.
2
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(I)-envelopes of unit balls and James' characterization of reflexivity

100%
Kalenda O.
Studia Mathematica
|
2007
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tom 182
|
nr 1
29-40
EN
We study the (I)-envelopes of the unit balls of Banach spaces. We show, in particular, that any nonreflexive space can be renormed in such a way that the (I)-envelope of the unit ball is not the whole bidual unit ball. Further, we give a simpler proof of James' characterization of reflexivity in the nonseparable case. We also study the spaces in which the (I)-envelope of the unit ball adds nothing.
3
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Quantification of the reciprocal Dunford-Pettis property

64%
Kalenda O., Spurný J.
Studia Mathematica
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2012
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tom 210
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nr 3
261-278
EN
We prove in particular that Banach spaces of the form C₀(Ω), where Ω is a locally compact space, enjoy a quantitative version of the reciprocal Dunford-Pettis property.
4
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Note on Bessaga-Klee classification

64%
Cúth M., Kalenda O.
Colloquium Mathematicum
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2015
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tom 140
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nr 1
59-74
EN
We collect several variants of the proof of the third case of the Bessaga-Klee relative classification of closed convex bodies in topological vector spaces. We were motivated by the fact that we have not found anywhere in the literature a complete correct proof. In particular, we point out an error in the proof given in the book of C. Bessaga and A. Pełczyński (1975). We further provide a simplified version of T. Dobrowolski's proof of the smooth classification of smooth convex bodies in Banach spaces which also works in the topological case.
5
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Baire classes of affine vector-valued functions

64%
Kalenda O., Spurný J.
Studia Mathematica
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2016
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tom 233
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nr 3
227-277
EN
We investigate Baire classes of strongly affine mappings with values in Fréchet spaces. We show, in particular, that the validity of the vector-valued Mokobodzki result on affine functions of the first Baire class is related to the approximation property of the range space. We further extend several results known for scalar functions on Choquet simplices or on dual balls of L₁-preduals to the vector-valued case. This concerns, in particular, affine classes of strongly affine Baire mappings, the abstract Dirichlet problem and the weak Dirichlet problem for Baire mappings. Some of these results have weaker conclusions than their scalar versions. We also establish an affine version of the Jayne-Rogers selection theorem.
6
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Fiber orders and compact spaces of uncountable weight

64%
Avilés A., Kalenda O.
Fundamenta Mathematicae
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2009
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tom 204
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nr 1
7-38
EN
We study an order relation on the fibers of a continuous map and its application to the study of the structure of compact spaces of uncountable weight.
7
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C(K) spaces which cannot be uniformly embedded into c₀(Γ)

51%
Pelant J., Holický P., Kalenda O.
Fundamenta Mathematicae
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2006
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tom 192
|
nr 3
245-254
EN
We give two examples of scattered compact spaces K such that C(K) is not uniformly homeomorphic to any subset of c₀(Γ) for any set Γ. The first one is [0,ω₁] and hence it has the smallest possible cardinality, the other one has the smallest possible height ω₀ + 1.
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