Hahn-Banach Theorem Research Articles

Applications of the Hahn-Banach Theorem, a Solution of the Moment Problem and the Related Approximation

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.

Analytical method for solving linear abstract differential equations of order [formula omitted]

In this paper, the concept of point-wise independent set of closed operators is introduced. The new concept together with the use of Hahn–Banach Theorem enable us to reduce an inhomogeneous linear abstract differential equation of order n namely, Anu(n)(t)+An−1u(n−1)(t)+⋯+A1u′(t)+A∘u(t)=f(t), into an inhomogeneous linear ordinary differential equation with constant coefficients of order n. The involved operators in the main abstract equation are assumed to be closed and linear on a given Banach space while the unknown function is assumed to be n-times differentiable.

Entropy and Thermodynamic Temperature in Nonequilibrium Classical Thermodynamics as Immediate Consequences of the Hahn–Banach Theorem: I. Existence

The Kelvin–Planck statement of the second law of thermodynamics is a stricture on the nature of heat receipt by any body suffering a cyclic process. It makes no mention of temperature or of entropy. Beginning with a Kelvin–Planck statement of the Second Law, we show that entropy and temperature—in particular, existence of functions that relate the local specific entropy and thermodynamic temperature to the local state in a material body—emerge immediately and simultaneously as consequences of the Hahn–Banach theorem. The existence of such functions of state requires no stipulation that their domains be restricted to equilibrium states. Further properties, including uniqueness, are addressed in a companion paper.

Entropy and Thermodynamic Temperature in Nonequilibrium Classical Thermodynamics as Immediate Consequences of the Hahn–Banach Theorem: II Properties

In a companion article it was shown in a certain precise sense that, for any thermodynamical theory that respects the Kelvin–Planck second law, the Hahn–Banach theorem immediately ensures the existence of a pair of continuous functions of the local material state—a specific entropy (entropy per mass) and a thermodynamic temperature—that together satisfy the Clausius–Duhem inequality for every process. There was no requirement that the local states considered be states of equilibrium. This article addresses questions about properties of the entropy and thermodynamic temperature functions so obtained: To what extent do such temperature functions provide a faithful reflection of “hotness”? In precisely which Kelvin–Planck theories is such a temperature function essentially unique, and, among those theories, for which is the entropy function also essentially unique? What is a thermometer for a Kelvin–Planck theory, and, for the theory, what properties does the existence of a thermometer confer? In all of these questions, the Hahn–Banach Theorem again plays a crucial role.

REFLEXIVITY OF LINEAR n-NORMED SPACE WITH RESPECT TO b-LINEAR FUNCTIONAL

In this paper, we discuss various consequences of Hahn-Banach theorem for bounded b-linear functional in linear n-normed space and describe the notion of re exivity of linear n-normed space with respect to bounded b-linear functional. The concepts of strong convergence and weak convergence of a sequence of vectors with respect to bounded b-linear functionals in linear n-normed space have been introduced and some of their properties are being discussed.

The equivalency of the Banach-Alaoglu theorem and the axiom of choice

The Axiom of Choice can be used to prove the Banach-Alaoglu theorem, while a weakened form of the Axiom of Choice is required to prove the full strength of the Hahn-Banach theorem, which is equivalent to the Banach-Alaoglu theorem. Although the Banach-Alaoglu theorem and the full- strength Axiom of Choice are not exactly equivalent, they are closely related.
 The Banach-Alaoglu theorem is a compactness theorem whose proof mainly depends on Tychonoff's theorem. In this article, Banach-Alaoglu theorem is equivalent to the axiom of choice, has been proved.

On the Equivalence in ZF+BPI of the Hahn–Banach Theorem and Three Classical Theorems

On the Equivalence in ZF+BPI of the Hahn–Banach Theorem and Three Classical Theorems

Further study on second order nonlocal problems monitored by an operator

In this note we prove the existence of mild solutions for nonlocal problems governed by semilinear second order differential inclusions which involves a nonlinear term driven by an operator. A first result is obtained in suitable Banach spaces in the lack of compactness both on the fundamental operator, generated by the linear part, and on the nonlinear multivalued term. This purpose is achieved by combining a fixed point theorem, a selection theorem and a containment theorem. Further we provide another existence result in reflexive spaces by using the classical Hahn–Banach theorem and a new selection proposition, proved here, for a multimap guided by an operator. This setting allows us to remove some assumptions required in the previous existence theorem. As a consequence of this last result we obtain the controllability of a problem driven by a wave equation on which an appropriate perturbation acts.

Convexity, Markov Operators, Approximation, and Related Optimization

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.

Extension of fuzzy linear operators on fuzzy Riesz spaces

Let E be a fuzzy order vector space, (F,v) be a fuzzy Riesz space with F fuzzy Dedekind complete, K⊂E is a non-empty convex subset, we show that for every fuzzy sublinear θ:K→F, there exists a fuzzy linear operator T:E→F such that T≤θ. This is the generalization of Hahn-Banach theorem in case where the range space is a fuzzy Dedekind complete fuzzy Riesz space. Related extension problems are also studied. Let K be a fuzzy Riesz subspace of E, T:K→F be a fuzzy positive operator, we give a characterization of extreme point of ε(T). We also prove that if K is a fuzzy majorizing vector subspace of E, and T:K→F is a fuzzy positive operator, then the convex set ε(T) has an extreme point.

G-Fuzzy Operator Norm

In our previous paper, it is shown that topology of [Formula: see text]-fuzzy normed linear space is generated by two types of open balls: one is elliptic and the other is circular. In the theoretical aspect of functional analysis, will this type of exception happen or not? To address this problem in this paper, firstly, [Formula: see text]-fuzzy bounded linear operators as well as [Formula: see text]-fuzzy bounded linear functionals are defined which are the key elements of functional analysis. Then, operator [Formula: see text]-fuzzy norms are introduced for both the cases using the idea of quasi-[Formula: see text]-norm family. The definition of operator [Formula: see text]-fuzzy norm is quite different from the existing operator fuzzy norm. Completeness of operator [Formula: see text]-fuzzy norm is investigated. Lastly, Hahn-Banach theorem in [Formula: see text]-fuzzy setting is studied using all the above concepts.

On Hahn-Banach theorem and some of its applications

Abstract First, this work provides an overview of some of the Hahn-Banach type theorems. Of note, some of these extension results for linear operators found recent applications to isotonicity of convex operators on a convex cone. Next, the work investigates applications of the Krein-Milman theorem to representation theory and elements of Choquet theory. A sandwich theorem of intercalating an affine function h h between f f and g , g, where f f\hspace{.25em} and – g \mbox{--}g are convex, f ≤ g f\le g on a finite-simplicial set, is recalled. Its recent topological version is also noted. Here, the novelty is that a finite-simplicial set may be unbounded in any locally convex topology on the domain space. Third, the paper summarizes and comments on recently published applications of a Hahn-Banach extension result for positive linear operators, combined with polynomial approximation on unbounded subsets, to the Markov moment problem. Some applications to concrete spaces are detailed as well. Finally, this work provides a characterization of a finite-dimensional convex bounded subset in terms of the property that any convex function defined on that subset is bounded below. This last property remains valid for a large class of convex operators.

Classical vs. non-Archimedean analysis: an approach via algebraic genericity

In this paper, we show new results and improvements of the non-Archimedean counterpart of classical analysis in the theory of lineability. Besides analyzing the algebraic genericity of sets of functions having properties regarding continuity, discontinuity, Lipschitzianity, differentiability and analyticity, we also study the lineability of sets of sequences having properties concerning boundedness and convergence. In particular we show (among several other results) the algebraic genericity of: (i) functions that do not satisfy Liouville’s theorem, (ii) sequences that do not satisfy the classical theorem of Cèsaro, or (iii) functionals that do not satisfy the classical Hahn–Banach theorem.

Hahn-Banach type theorems and the separation of convex sets for fuzzy quasi-normed spaces

<abstract> <p>In this paper, we first study continuous linear functionals on a fuzzy quasi-normed space, obtain a characterization of continuous linear functionals, and point out that the set of all continuous linear functionals forms a convex cone and can be equipped with a weak fuzzy quasi-norm. Next, we prove a theorem of Hahn-Banach type and two separation theorems for convex subsets of fuzzy quasinormed spaces.</p> </abstract>

Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras

We establish several deep existence criteria for conditional expectations on von Neumann algebras, and then apply this theory to develop a noncommutative theory of representing measures of characters of a function algebra. Our main cycle of results describes what may be understood as a ‘noncommutative Hoffman-Rossi theorem’ giving the existence of weak* continuous ‘noncommutative representing measures’ for so-called D-characters. These results may also be viewed as ‘module’ Hahn-Banach extension theorems for weak* continuous ‘characters’ into possibly noninjective von Neumann algebras. In closing we introduce the notion of ‘noncommutative Jensen measures’, and show that as in the classical case representing measures of logmodular algebras are Jensen measures. The proofs of the two main cycles of results rely on the delicate interplay of Tomita-Takesaki theory, noncommutative Radon-Nikodym derivatives, Connes cocycles, Haagerup noncommutative Lp-spaces, Haagerup's reduction theorem, etc.

Representation of uniform boundedness principle and Hahn–Banach theorem in linear n-normed space

The concept of b-linear functional and its different types of continuity in linear n-normed space are presented and some of their properties are being established. We derive the Uniform Boundedness Principle and Hahn-Banach extension Theorem with the help of bounded b-linear functionals in the case of linear n-normed spaces and discuss some examples and applications. Finally, we present the concept of weak*convergence for the sequence of bounded b-linear functionals in linear n-normed space.

On the pillars of Functional Analysis

Many authors consider that the main pillars of Functional Analysis are the Hahn–Banach Theorem, the Uniform Boundedness Principle and the Open Mapping Principle. The first one is derived from Zorn’s Lemma, while the latter two usually are obtained from Baire’s Category Theorem. In this paper we show that these three pillars should be either just two or at least eight, since the Uniform Boundedness Principle, the Open Mapping Principle and another five theorems are equivalent, as we show in a very elemental way. Since one can give an almost trivial proof of the Uniform Boundedness Principle that does not require the Baire’s theorem, we conclude that this is also the case for the other equivalent theorems that, in this way, are simultaneously proved in a simple, brief and concise way that sheds light on their nature.

Order Versions of the Hahn–Banach Theorem and Envelopes. II. Applications to Function Theory

Order Versions of the Hahn–Banach Theorem and Envelopes. II. Applications to Function Theory

Banach-Steinhaus Theorem for the Space P of All Primitives of Henstock-Kurzweil Integrable Functions

In this paper, it is shown how the Banach-Steinhaus theorem for the space P of all primitives of Henstock-Kurzweil integrable functions on a closed bounded interval, equipped with the uniform norm, can follow from the Banach-Steinhaus theorem for the Denjoy space by applying the classical Hahn-Banach theorem and Riesz representation theorem.

Convex geometry of Markovian Lindblad dynamics and witnessing non-Markovianity

We develop a theory of linear witnesses for detecting non-Markovianity, based on the geometric structure of the set of Choi states for all Markovian evolutions having Lindblad-type generators. We show that the set of all such Markovian Choi states form a convex and compact set under the small time interval approximation. Invoking geometric Hahn–Banach theorem, we construct linear witnesses to separate a given non-Markovian Choi state from the set of Markovian Choi states. We present examples of such witnesses for dephasing channel and Pauli channel in case of qubits. Furthermore, we have devised another method of detection NM of various qubit channels, by using projective measurements in the Bell basis. This can be done without the knowledge of what specific kind of channel we are dealing with. This gives us a huge operational advantage for NM detection. We further investigate the geometric structure of the Markovian Choi states to find that they do not form a polytope. This presents a platform to consider nonlinear improvement of non-Markovianity witnesses.