Hahn-Banach Extension Theorem Research Articles

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.

From the Hahn–Banach extension theorem to the isotonicity of convex functions and the majorization theory

The property of isotonicity of a continuous convex function on the positive cone is characterized via subdifferentials. This is used to illustrate a new generalization of the Hardy–Littlewood–Polya inequality of majorization to the case of functions of a vector variable.

Amenability and Hahn-Banach extension property for set-valued mappings

Amenability is an important notion in harmonic analysis on groups and semigroups, and their associated Banach algebras. In this paper, we present some characterization of a semitopological semigroup $S$ on the existence of a left invariant mean on $\text{\rm LUC}(S)$, $\text{\rm AP}(S)$ and $\text{\rm WAP}(S)$ in terms of Hahn-Banach extension theorem, which extend the first author's early results in 1970s. Moreover, we refine and extend the well known Day's result and Mitchell's results on fixed point properties for set-valued mappings. As an application, we give an application of our result to a class of Banach algebras related to amenability of groups and semigroups.

Hahn-Banach extension theorem for interval-valued functions and existence of quasilinear functionals

Hahn-Banach extension theorem for interval-valued functions and existence of quasilinear functionals

Amenability and Fan–Glicksberg theorem for set-valued mappings

In this paper, we begin by discussion of some well known results on the existence of left invariant means in the spaces: LUC(S), AP(S) and W AP(S) with Hahn-Banach extension theorem. We then give a new and precise proof of the well known Fan–Glicksberg fixed point theorem. This is then followed by a discussion on some related open problems.

Global solutions in the critical Besov space for the non-cutoff Boltzmann equation

The Boltzmann equation is studied without the cutoff assumption. Under a perturbative setting, a unique global solution of the Cauchy problem of the equation is established in a critical Chemin–Lerner space. In order to analyze the collisional term of the equation, a Chemin–Lerner norm is combined with a non-isotropic norm with respect to a velocity variable, which yields an a priori estimate for an energy estimate. Together with local existence following from commutator estimates and the Hahn–Banach extension theorem, the desired solution is obtained.

Subdifferential calculus for invariant linear ordered vector space-valued operators and applications

We give a direct proof of sandwich-type theorems for linear invariant partially ordered vector space operators in the setting of convexity. As consequences, we deduce equivalence results between sandwich, Hahn-Banach, separation and Krein-type extension theorems, Fenchel duality, Farkas and Kuhn-Tucker-type minimization results and subdifferential formulas in the context of invariance. As applications, we give Tarski-type extension theorems and related examples for vector lattice-valued invariant probabilities, defined on suitable kinds of events.

The Hahn–Banach theorem almost everywhere

The aim of this work is to present an almost everywhere version of the Hahn–Banach extension theorem.

Dual Spaces and Hahn-Banach Theorem

Summary In this article, we deal with dual spaces and the Hahn-Banach Theorem. At the first, we defined dual spaces of real linear spaces and proved related basic properties. Next, we defined dual spaces of real normed spaces. We formed the definitions based on dual spaces of real linear spaces. In addition, we proved properties of the norm about elements of dual spaces. For the proof we referred to descriptions in the article [21]. Finally, applying theorems of the second section, we proved the Hahn-Banach extension theorem in real normed spaces. We have used extensively used [17].

The Hahn-Banach Extension Theorem for Fuzzy Normed Spaces Revisited

This paper deals with fuzzy normed spaces in the sense of Cheng and Mordeson. We characterize fuzzy norms in terms of ascending and separating families of seminorms and prove an extension theorem for continuous linear functionals on a fuzzy normed space. Our result generalizes the classical Hahn-Banach extension theorem for normed spaces.

A FUZZY VERSION OF HAHN-BANACH EXTENSION THEOREM

In this paper, a fuzzy version of the analytic form of Hahn-Banachextension theorem is given. As application, the Hahn-Banach theorem for$r$-fuzzy bounded linear functionals on $r$-fuzzy normedlinear spaces is obtained.

Some new versions of Hahn-Banach theorem

In 1997, L.-G. Huang and X. Zhang introduced the cone metric space, which is the generalization of the metric space. Further, some authors introduced the cone normed space. In this paper, we introduce the conceptions of bounded linear operator from a normed space to a cone normed space and bounded linear operator between two cone normed spaces. A general Hahn-Banach type linear operator extension theorem from a linear space to a partial ordering linear space, with the order is induced by a minimal cone, is proven. Two preserved-norm bounded linear operator extension theorems, one is from a normed space to a cone normed space and another is between two cone normed spaces, and some corollaries are given.

About Hahn–Banach extension theorems and applications to set-valued optimization

In this paper we revise several generalizations of the algebraic Hanh–Banach extension theorem for set-valued maps and give new versions in terms of set-relations and selections. We apply our results by establishing optimality conditions for a set-valued constrained optimization problem considering several types of solutions. Finally, we give necessary conditions for the existence of affine selections via affinelike maps in the framework of linear spaces.

A geometric form of the Hahn-Banach extension theorem for <italic>L</italic><sup>0</sup>-linear func-tions and the Goldstine-Weston theorem in random normed modules

In this paper, we present a geometric form of the Hahn-Banach extension theorem for <italic>L</italic>⁰-linear functions and prove that the geometric form is equivalent to the algebraic form of the Hahn-Banach extension theorem. Further, we use the geometric form to give a new and simpler proof of a known basic strict separation theorem in random locally convex modules. Finally, using the basic strict separation theorem we establish the Goldstine-Weston theorem in random normed modules under the two kinds of topologies|the (<italic>ε, λ</italic>)-topology and the locally <italic>L</italic>⁰-convex topology, while we also provide a counterexample showing that the Goldstine-Weston theorem under the locally <italic>L</italic>⁰-convex topology may not hold for random normed modules without the countable concatenation property.

Hahn-Banach extension theorems over the space of fuzzy elements

The conventional Hahn-Banach extension theorem over a vector space has been widely used to derive many important and interesting results in nonlinear analysis, vector optimization and mathematical economics. Although the space of fuzzy elements is not a real vector space, the Hahn-Banach extension theorems over the space of fuzzy elements and the nonstandard normed space of fuzzy elements are presented in this paper. The work also shows the possible applications of the fuzzy-valued problems to nonlinear analysis, vector optimization and mathematical economics.

Relations between some basic results derived from two kinds of topologies for a random locally convex module

The purpose of this paper is to exhibit the relations between some basic results derived from the two kinds of topologies (namely the ( ε , λ ) -topology and the stronger locally L 0 -convex topology) for a random locally convex module. First, we give an extremely simple proof of the known Hahn–Banach extension theorem for L 0 -linear functions as well as its continuous variant. Then we give the relations between the hyperplane separation theorems in [D. Filipović, M. Kupper, N. Vogelpoth, Separation and duality in locally L 0 -convex modules, J. Funct. Anal. 256 (2009) 3996–4029] and a basic strict separation theorem in [T.X. Guo, H.X. Xiao, X.X. Chen, A basic strict separation theorem in random locally convex modules, Nonlinear Anal. 71 (2009) 3794–3804]: in the process we also obtain a very useful fact that a random locally convex module with the countable concatenation property must have the same completeness under the two topologies. As applications of the fact, we prove that most of the previously established principal results of random conjugate spaces of random normed modules under the ( ε , λ ) -topology are still valid under the locally L 0 -convex topology, which considerably enriches financial applications of random normed modules.

How Incomputable Is the Separable Hahn-Banach Theorem?

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second-order arithmetic WKL0. We study analogies and differences between WKL0 and the class of Sep-computable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the Hahn-Banach Extension Theorem is Sep-complete.

Hahn–Banach theorems in nonstandard normed interval spaces

The conventional Hahn–Banach extension theorem based on vector space has been widely used to obtain many important and interesting results in nonlinear analysis, vector optimization and mathematical economics. Although the interval space is not a real vector space, the Hahn–Banach extension theorems based on interval spaces and nonstandard normed interval spaces can still be derived in this paper, which also shows the possible applications by considering the interval-valued problems in nonlinear analysis, vector optimization and mathematical economics.

Weak subdifferential for set-valued mappings and its applications

In this paper, the existence theorems of two kinds of weak subgradients for set-valued mappings, which are the generalizations of Theorem 7 in [G.Y. Chen, J. Jahn, Optimality conditions for set-valued optimization problems, Math. Methods Oper. Res. 48 (2) (1998) 187–200] and Theorem 4.1 in [J.W. Peng, H.W.J. Lee, W.D. Rong, X.M. Yang, Hahn–Banach theorems and subgradients of set-valued maps, Math. Methods Oper. Res. 61 (2005) 281–297], respectively, are proved by virtue of a Hahn–Banach extension theorem. Moreover, some properties of the weak subdifferential for set-valued mappings are obtained by using a so-called Sandwich theorem. Finally, necessary and sufficient optimality conditions are discussed for set-valued optimization problems, whose constraint sets are determined by a fixed set and a set-valued mapping, respectively.