Extreme points of unbounded, closed and convex sets in Banach spaces

1984 ◽  
Vol 95 (2) ◽  
pp. 319-323 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Sets ◽  
Extreme Points ◽  
Convex Subsets

In this paper we examine the existence of extreme points in unbounded, closed and convex subsets K of a Banach space X.

scholarly journals Strongly exposed points in bases for the positive cone of ordered Banach spaces and characterizations of l1(Г)

1986 ◽  
Vol 29 (2) ◽  
pp. 271-282 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Sets ◽  
Extreme Points ◽  
Positive Cone ◽  
Infinite Dimensional ◽  
Strongly Exposed Points ◽  
Nikodym Property ◽  
Choquet Theory ◽  
Ordered Banach Spaces

The study of extreme, strongly exposed points of closed, convex and bounded sets in Banach spaces has been developed especially by the interconnection of the Radon–Nikodým property with the geometry of closed, convex and bounded subsets of Banach spaces [5],[2] . In the theory of ordered Banach spaces as well as in the Choquet theory, [4], we are interested in the study of a special type of convex sets, not necessarily bounded, namely the bases for the positive cone. In [7] the geometry (extreme points, dentability) of closed and convex subsets K of a Banach space X with the Radon-Nikodým property is studied and special emphasis has been given to the case where K is a base for acone P of X. In [6, Theorem 1], it is proved that an infinite-dimensional, separable, locally solid lattice Banach space is order-isomorphic to l1 if, and only if, X has the Krein–Milman property and its positive cone has a bounded base.

scholarly journals Mazur's intersection property of balls for compact convex sets

1987 ◽  
Vol 35 (2) ◽  
pp. 267-274 ◽  
Author(s):  
J. H. M. Whitfield ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Intersection Property ◽  
Compact Sets ◽  
Compact Convex Set ◽  
Mazur's Intersection Property

We show that every compact convex set in a Banach space X is an intersection of balls provided the cone generated by the set of all extreme points of the dual unit ball of X* is dense in X* in the topology of uniform convergence on compact sets in X. This allows us to renorm every Banach space with transfinite Schauder basis by a norm which shares the mentioned intersection property.

The Structure of C*-Convex Sets

1994 ◽  
Vol 46 (5) ◽  
pp. 1007-1026 ◽  
Author(s):  
Phillip B. Morenz
Keyword(s):  
Convex Sets ◽  
Extreme Points ◽  
Structural Elements ◽  
The Right ◽  
The Relationship ◽  
Carathéodory Theorem ◽  
Convex Subsets

AbstractCompact C*-convex subsets of Mn correspond exactly to n-th matrix ranges of operators. The main result of this paper is to discover the “right” analog of linear extreme points, called structural elements, and then to prove a generalised Krein-Milman theorem for C*-convex subsets of Mn. The relationship between structural elements and an earlier attempted generalisation, called C*-extreme points, is examined, solving affirmatively a conjecture of Loebl and Paulsen [8]. An improved bound for a C* -convex version of the Caratheodory theorem for convex sets is also given.

EXTREME POINTS FOR COMBINATORIAL BANACH SPACES

2018 ◽  
Vol 61 (2) ◽  
pp. 487-500
Author(s):  
KEVIN BEANLAND ◽  
NOAH DUNCAN ◽  
MICHAEL HOLT ◽  
JAMES QUIGLEY
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Unit Ball ◽  
Banach Spaces ◽  
Extreme Points ◽  
Explicit Construction ◽  
Stability Properties ◽  
Original Space ◽  
Regular Family

AbstractA norm ‖ċ‖ on c00 is called combinatorial if there is a regular family of finite subsets $\mathcal{F}$, so that $\|x\|=\sup_{F \in \mathcal{F}} \sum_{i \in F} |x(i)|$. We prove the set of extreme points of the ball of a combinatorial Banach space is countable. This extends a theorem of Shura and Trautman. The second contribution of this article is to exhibit many new examples of extreme points for the unit ball of dual Tsirelson's original space and give an explicit construction of an uncountable collection of extreme points of the ball of Tsirelson's original space. We also prove some stability properties of the intermediate norms used to define Tsirelson's space and give a lower bound of the stabilization function for these intermediate norms.

scholarly journals The Problem of Image Recovery by the Metric Projections in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yasunori Kimura ◽  
Kazuhide Nakajo
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Countable Family ◽  
Iterative Sequence ◽  
Image Recovery ◽  
Convergence Theorems ◽  
Metric Projections ◽  
Convex Subsets

We consider the problem of image recovery by the metric projections in a real Banach space. For a countable family of nonempty closed convex subsets, we generate an iterative sequence converging weakly to a point in the intersection of these subsets. Our convergence theorems extend the results proved by Bregman and Crombez.

Some more characterizations of Banach spaces containing l1

1976 ◽  
Vol 80 (2) ◽  
pp. 269-276 ◽  
Author(s):  
Richard Haydon
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Banach Spaces ◽  
Convex Subset ◽  
Compact Convex ◽  
Extreme Points ◽  
Compact Convex Subset ◽  
Closed Convex Hull ◽  
Separability Assumption

In a series of recent papers ((10), (9) and (11)) Rosenthal and Odell have given a number of characterizations of Banach spaces that contain subspaces isomorphic (that is, linearly homeomorphic) to the space l1 of absolutely summable series. The methods of (9) and (11) are applicable only in the case of separable Banach spaces and some of the results there were established only in this case. We demonstrate here, without the separability assumption, one of these characterizations:a Banach space B contains no subspace isomorphic to l1 if and only if every weak* compact convex subset of B* is the norm closed convex hull of its extreme points.

scholarly journals Extreme integral polynomials on a complex Banach space

2003 ◽  
Vol 92 (1) ◽  
pp. 129 ◽  
Author(s):  
Seán Dineen
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Complex Space ◽  
Extreme Points ◽  
Complex Banach Space ◽  
The Real

We obtain upper and lower set-theoretic inclusion estimates for the set of extreme points of the unit balls of $\mathcal{P}_{I}({}^{n}\!E)$ and $\mathcal{P}_{N}({}^{n}\!E)$, the spaces of $n$-homogeneous integral and nuclear polynomials, respectively, on a complex Banach space $E$. For certain collections of Banach spaces we fully characterise these extreme points. Our results show a difference between the real and complex space cases.

scholarly journals Strong convergence theorems for equilibrium problems involving Bregman functions in Banach spaces

2017 ◽  
Vol 48 (2) ◽  
pp. 159-184
Author(s):  
Eskandar Naraghirad ◽  
Sara Timnak
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Equilibrium Problem ◽  
Banach Spaces ◽  
Convex Sets ◽  
Iterative Algorithms ◽  
Iterative Sequence ◽  
Bregman Projection ◽  
Practical Calculation ◽  
Bregman Functions

In this paper, using Bregman functions, we introduce new Halpern-type iterative algorithms for finding a solution of an equilibrium problem in Banach spaces. We prove the strong convergence of a modified Halpern-type scheme to an element of the set of solution of an equilibrium problem in a reflexive Banach space. This scheme has an advantage that we do not use any Bregman projection of a point on the intersection of closed and convex sets in a practical calculation of the iterative sequence. Finally, some application of our results to the problem of finding a minimizer of a continuously Fr\'{e}chet differentiable and convex function in a Banach space is presented. Our results improve and generalize many known results in the current literature.

scholarly journals Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 150
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Dimensional Algebra ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Linear Subspaces ◽  
Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

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