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Search Results (4)

Search Parameters:
Keywords = Krein–Milman

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18 pages, 354 KiB  
Review
Applications of the Hahn-Banach Theorem, a Solution of the Moment Problem and the Related Approximation
by Octav Olteanu
Mathematics 2024, 12(18), 2878; https://doi.org/10.3390/math12182878 - 15 Sep 2024
Cited by 1 | Viewed by 1076
Abstract
We start by an application the of Krein–Milman theorem to the integral representation of completely monotonic functions. Elements of convex optimization are also mentioned. The paper continues with applications of Hahn–Banach-type theorems and polynomial approximation to obtain recent results on the moment problem [...] Read more.
We start by an application the of Krein–Milman theorem to the integral representation of completely monotonic functions. Elements of convex optimization are also mentioned. The paper continues with applications of Hahn–Banach-type theorems and polynomial approximation to obtain recent results on the moment problem on the unbounded closed interval [0,+∞). Necessary and sufficient conditions for the existence and uniqueness of the solution are pointed out. Operator-valued moment problems and a scalar-valued moment problem are solved. Full article
(This article belongs to the Section E6: Functional Interpolation)
16 pages, 337 KiB  
Article
On Bishop–Phelps and Krein–Milman Properties
by Francisco Javier García-Pacheco
Mathematics 2023, 11(21), 4473; https://doi.org/10.3390/math11214473 - 28 Oct 2023
Cited by 1 | Viewed by 1434
Abstract
A real topological vector space is said to have the Krein–Milman property if every bounded, closed, convex subset has an extreme point. In the case of every bounded, closed, convex subset is the closed convex hull of its extreme points, then we say [...] Read more.
A real topological vector space is said to have the Krein–Milman property if every bounded, closed, convex subset has an extreme point. In the case of every bounded, closed, convex subset is the closed convex hull of its extreme points, then we say that the topological vector space satisfies the strong Krein–Milman property. The strong Krein–Milman property trivially implies the Krein–Milman property. We provide a sufficient condition for these two properties to be equivalent in the class of Hausdorff locally convex real topological vector spaces. This sufficient condition is the Bishop–Phelps property, which we introduce for real topological vector spaces by means of uniform convergence linear topologies. We study the inheritance of the Bishop–Phelps property. Nontrivial examples of topological vector spaces failing the Krein–Milman property are also given, providing us with necessary conditions to assure that the Krein–Milman property is satisfied. Finally, a sufficient condition to assure the Krein–Milman property is discussed. Full article
(This article belongs to the Collection Topology and Foundations)
17 pages, 342 KiB  
Review
Convexity, Markov Operators, Approximation, and Related Optimization
by Octav Olteanu
Mathematics 2022, 10(15), 2775; https://doi.org/10.3390/math10152775 - 4 Aug 2022
Cited by 3 | Viewed by 1748
Abstract
The present review paper provides recent results on convexity and its applications to the constrained extension of linear operators, motivated by the existence of subgradients of continuous convex operators, the Markov moment problem and related Markov operators, approximation using the Krein–Milman theorem, related [...] Read more.
The present review paper provides recent results on convexity and its applications to the constrained extension of linear operators, motivated by the existence of subgradients of continuous convex operators, the Markov moment problem and related Markov operators, approximation using the Krein–Milman theorem, related optimization, and polynomial approximation on unbounded subsets. In many cases, the Mazur–Orlicz theorem also leads to Markov operators as solutions. The common point of all these results is the Hahn–Banach theorem and its consequences, supplied by specific results in polynomial approximation. All these theorems or their proofs essentially involve the notion of convexity. Full article
(This article belongs to the Special Issue Ordered Topological Vector Spaces, Operators Acting on Them and Related Applications)
16 pages, 317 KiB  
Review
From Hahn–Banach Type Theorems to the Markov Moment Problem, Sandwich Theorems and Further Applications
by Octav Olteanu
Mathematics 2020, 8(8), 1328; https://doi.org/10.3390/math8081328 - 10 Aug 2020
Cited by 6 | Viewed by 2441
Abstract
The aim of this review paper is to recall known solutions for two Markov moment problems, which can be formulated as Hahn–Banach extension theorems, in order to emphasize their relationship with the following problems: (1) pointing out a previously published sandwich theorem of [...] Read more.
The aim of this review paper is to recall known solutions for two Markov moment problems, which can be formulated as Hahn–Banach extension theorems, in order to emphasize their relationship with the following problems: (1) pointing out a previously published sandwich theorem of the type f ≤ h ≤ g, where f, −g are convex functionals and h is an affine functional, over a finite-simplicial set X, and proving a topological version for this result; (2) characterizing isotonicity of convex operators over arbitrary convex cones; giving a sharp direct proof for one of the generalizations of Hahn–Banach theorem applied to the isotonicity; (3) extending inequalities assumed to be valid on a small subset, to the entire positive cone of the domain space, via Krein–Milman or Carathéodory’s theorem. Thus, we point out some earlier, as well as new applications of the Hahn–Banach type theorems, emphasizing the topological versions of these applications. Full article
(This article belongs to the Special Issue Hahn-Banach Theorem, Polynomial Approximation, Moment Problems, and Related Inverse Problems)
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