Existence of diametrically complete sets with empty interior in reflexive and separable Banach spaces

2020 ◽  
Vol 278 (7) ◽  
pp. 108418
Author(s):  
Monika Budzyńska ◽  
Tadeusz Kuczumow ◽  
Simeon Reich ◽  
Mariola Walczyk
Keyword(s):  
Banach Spaces ◽  
Empty Interior ◽  
Complete Sets

Schauder bases and diametrically complete sets with empty interior

2018 ◽  
Vol 463 (2) ◽  
pp. 452-460 ◽  
Author(s):  
Monika Budzyńska ◽  
Aleksandra Grzesik ◽  
Wiesława Kaczor ◽  
Tadeusz Kuczumow
Keyword(s):  
Empty Interior ◽  
Schauder Bases ◽  
Complete Sets

Cone-Monotone Functions: Differentiability and Continuity

2005 ◽  
Vol 57 (5) ◽  
pp. 961-982 ◽  
Author(s):  
Jonathan M. Borwein ◽  
Xianfu Wang
Keyword(s):  
Banach Spaces ◽  
Convex Cone ◽  
Continuous Functions ◽  
Monotone Functions ◽  
Empty Interior ◽  
Space Of Continuous Functions

AbstractWe provide a porosity-based approach to the differentiability and continuity of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone K with non-empty interior. We also show that the set of nowhere K-monotone functions has a σ-porous complement in the space of continuous functions endowed with the uniform metric.

scholarly journals Complete sets and completion of sets in Banach spaces

2013 ◽  
Vol 174 (4) ◽  
pp. 587-597 ◽  
Author(s):  
Horst Martini ◽  
Pier Luigi Papini ◽  
Margarita Spirova
Keyword(s):  
Banach Spaces ◽  
Complete Sets

Diametrically Maximal and Constant Width Sets in Banach Spaces

2006 ◽  
Vol 58 (4) ◽  
pp. 820-842 ◽  
Author(s):  
J. P. Moreno ◽  
P. L. Papini ◽  
R. R. Phelps
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Hausdorff Space ◽  
Constant Width ◽  
Negative Answer ◽  
Empty Interior ◽  
Finite Dimensional ◽  
Jung Constant

AbstractWe characterize diametrically maximal and constant width sets inC(K), whereKis any compact Hausdorff space. These results are applied to prove that the sum of two diametrically maximal sets needs not be diametrically maximal, thus solving a question raised in a paper by Groemer. A characterization of diametrically maximal sets inis also given, providing a negative answer to Groemer's problem in finite dimensional spaces. We characterize constant width sets inc0(I), for everyI, and then we establish the connections between the Jung constant of a Banach space and the existence of constant width sets with empty interior. Porosity properties of families of sets of constant width and rotundity properties of diametrically maximal sets are also investigated. Finally, we present some results concerning non-reflexive and Hilbert spaces.

Constructions of complete sets

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
Pier Luigi Papini ◽  
Senlin Wu
Keyword(s):  
Banach Spaces ◽  
Constant Width ◽  
General Context ◽  
Easy Task ◽  
New Construction ◽  
Complete Sets

AbstractConstructing constant width sets or, more generally, complete sets in Banach spaces seems to be a not so easy task. A new construction working in separable Banach spaces is presented, and Bavaud’s and Lachand-Robert and Oudet’s constructions of complete sets are extended to a more general context.

Weyl Spectral Identity and Biquasitriangularity

2015 ◽  
Vol 59 (2) ◽  
pp. 363-375 ◽  
Author(s):  
C. S. Kubrusly ◽  
B. P. Duggal
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Empty Interior ◽  
Infinite Dimensional ◽  
Weyl Spectrum ◽  
Spectral Identity

AbstractLet A and B be operators acting on infinite-dimensional complex Banach spaces. We say that the Weyl spectral identity holds for the tensor product A⊗B if σw(A⊗B) = σw(A)·σ(B)∪σ(A)·σw(B), where σ(·) and σw(·) stand for the spectrum and the Weyl spectrum, respectively. Conditions on A and B for which the Weyl spectral identity holds are investigated. Especially, it is shown that if A and B are biquasitriangular (in particular, if the spectra of A and B have empty interior), then the Weyl spectral identity holds. It is also proved that if A and B are biquasitriangular, then the tensor product A ⊗ B is biquasitriangular.

scholarly journals Generalized Uniqueness Theorem for Ordinary Differential Equations in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Ezzat R. Hassan ◽  
M. Sh. Alhuthali ◽  
M. M. Al-Ghanmi
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Uniqueness Theorem ◽  
Banach Spaces ◽  
Nonlinear Ordinary Differential Equations ◽  
Empty Interior ◽  
Reflexive Banach Spaces ◽  
Uniqueness Criterion ◽  
The Cauchy Problem

We consider nonlinear ordinary differential equations in Banach spaces. Uniqueness criterion for the Cauchy problem is given when any of the standard dissipative-type conditions does apply. A similar scalar result has been studied by Majorana (1991). Useful examples of reflexive Banach spaces whose positive cones have empty interior has been given as well.

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