Search

We prove a special case of the Banach-Steinhaus theorem for Sargent spaces, making use of the Hahn-Banach theorem and by-passing the proof by Sargent which requires the Baire category theorem. [ABSTRACT FROM AUTHOR]

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 reexivity 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. [ABSTRACT FROM AUTHOR]

The presented paper is a compendium or a kind of précis of relationships between the four classical theorems of mathematical analysis. More precisely, the first aim of this paper is to present the equivalence of the four classical theorems: the Hahn–Banach theorem, the Mazur–Orlicz theorem, the Markov–Kakutani fixed-point theorem and the von Neumann theorem on amenability of Abelian groups. The second purpose is to prove these equivalences in Zermelo–Fraenkel set theory with the axiom of choice replaced by the Boolean prime ideal theorem, from which we can also prove the Hahn–Banach theorem. [ABSTRACT FROM AUTHOR]

Let L² ((0; 1); x4 dx) be the weighted Lebesgue space of all measurable functions f : (0; 1) → C, satisfying ∫0¹ t4 |f(t)|² dt < +∞. Let J−5/2 be the Bessel function of the first kind of index −5/2 and (ρk)k∈N be a sequence of distinct nonzero complex numbers. Necessary and sufficient conditions for the completeness of the system {ρk² √xρk J−5/2(xρk) : k ∈ N} in the space L² ((0; 1); x4 dx) are found in terms of an entire function with the set of zeros coinciding with the sequence (ρk)k∈N. In this case, we study an integral representation of some class E4,+ of even entire functions of exponential type σ ≤ 1. This complements similar results on approximation properties of the systems of Bessel functions of negative half-integer index less than −1, due to B. Vynnyts’kyi, V. Dilnyi, O. Shavala and the author. [ABSTRACT FROM AUTHOR]

In this paper, we give a multiplication operator representation of bounded self-adjoint operators T on a Hilbert space H such that { T k φ } k = 0 ∞ is a frame for H , for some φ ∈ H . We state a necessary condition in order for a frame { f k } k = 1 ∞ to have a representation of the form { T k f 1 } k = 0 ∞ . Frame sequence { T k φ } k = 0 ∞ with the synthesis operator U, have also been characterized in terms of the behavior spectrum of U ∗ U | N (U) ⊥ . Next, using the operator response of an element with respect to a unit vector in H , frames { f k } k = 1 ∞ of the form { T n f 1 } n = 0 ∞ are characterized. We also consider stability frames as { T k φ } k = 0 ∞ . Finally, we conclude this note by raising a conjecture connecting frame theory and operator theory. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

I show that de Finetti's coherence theorem is equivalent to the Hahn-Banach theorem and discuss some consequences of this result. First, the result unites two aspects of de Finetti's thought in a nice way: a corollary of the result is that the coherence theorem implies the existence of a fair countable lottery, which de Finetti appealed to in his arguments against countable additivity. Another corollary of the result is the existence of sets that are not Lebesgue measurable. I offer a subjectivist interpretation of this corollary that is concordant with de Finetti's views. I conclude by pointing out that my result shows that there is a sense in which de Finetti's theory of subjective probability is necessarily nonconstructive. This raises questions about whether the coherence theorem can underwrite a legitimate theory of rational belief. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem on the existence of the upper (lower) envelope of a convex cone or, more generally, a convex set for functions on the projective limit of vector lattices with values in the completion of the Kantorovich space or on the extended real line. We propose vectorial, ordinal, and topological dual interpretations of the existence conditions for such envelopes and the method of constructing it. Applications to the problem on the existence of a nontrivial (pluri)subharmonic and/or (pluri)harmonic minorant for functions in domains of a finite-dimensional real or complex space are considered. We also propose general approaches to problems on the nontriviality of weight classes of holomorphic functions, describing zero (sub)sets for such classes of holomorphic functions, and to the problem of representing a meromorphic function as a ratio of holomorphic function from a given weight class. [ABSTRACT FROM AUTHOR]

Our title means that the Knaster-Kuratowski Mazurkiewicz theorem in 1929 implies the Hahn-Banach theorem. This theorem originated from Hahn in 1926 and Banach in 1929 is of basic importance in the analysis of problems concerning the existence of continuous linear functionals. Its consequences and applications cover hundreds of papers. For a long period, some authors studied the relation of results of the HahnBanach theorem and the Brouwer xed point theorem in 1912 (equivalently, the KKM theorem in 1929). In the present article, we recall some history of such study, and show that the Hahn-Banach theorem can be derived from the KKM theorem and not conversely. Consequently, all consequences and applications of the Hahn-Banach theorem belong to a partial realm of the KKM theory.

In this article, we study an Euler-Bernoulli viscoelastic control system which is dissipative due to the presence of the viscoelastic term. The main feature which distinguishes this paper from other related works lies in the fact that we no longer impose traditional conditions such as complete monotonicity and decay property on the kernel function g. Without loss of generality, we study the system in the case of $g\equiv 1$. By means of the duality principle and the Hahn-Banach theorem, we show that the system with g=1 is approximately controllable in the appropriate function space.

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.

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. [ABSTRACT FROM AUTHOR]

We present here a simple and direct proof of the classic geometric version of the Hahn–Banach theorem from its analytic version, in the real case. The reverse implication, and the direct proofs of both versions, are well known. They are summarized here for reader's convenience. For the complex case, in both versions the Hahn–Banach Theorem is deduced from the real case, as is well known. [ABSTRACT FROM AUTHOR]

During half a century, singular traces on ideals of Hilbert space operators have been constructed by looking for linear forms on associated sequence ideals. Only recently, the author was able to eliminate this auxiliary step by directly applying Banach's version of the extension theorem; see (Integral Equ. Oper. Theory 91, 21, 2019 and 92, 7, 2020). Of course, the relationship between the new approach and the older ones must be investigated. In the first paper, this was done for L 1 , ∞ (H) . To save space, such considerations were postponed in the second paper, which deals with a large class of principal ideals, called simply generated. This omission will now be rectified. [ABSTRACT FROM AUTHOR]

In this paper, we develop the concept of dual and approximately dual frames in Banach spaces via semi-inner products and some properties of dual and approximate dual frames are investigated. Also, we introduce g-dual frames in these spaces and some relationships between g-duals and approximate duals are stated. Finally, the ε-nearly g-dual frames and their relations with g-duals are studied in Banach spaces using semi-inner products. [ABSTRACT FROM AUTHOR]

We present direct and self-contained proofs of the existence of singular traces on sufficiently large classes of operator ideals that live on the separable infinite-dimensional real Hilbert space. Our only tool is Banach’s version of the extension theorem of linear forms. Thanks to the use of dyadic representations of operators, all proofs have become straightforward. [ABSTRACT FROM AUTHOR]

Basic results for nonlinear operators are given. These results include nonlinear versions of the classical uniform boundedness theorem and Hahn-Banach theorem. Further, mappings of a metrizable space into another normed space can belong to some normed spaces if one defines suitable norms. [ABSTRACT FROM AUTHOR]

Abstract We introduce a notion of compatible quasi-ordered groups which unifies valued and ordered abelian groups. It was proved by S.M. Fakhruddin that a compatible quasi-order on a field is always either an order or a valuation. We show here that the group case is more complicated than the field case and describe the general structure of a compatible quasi-ordered abelian group. We then define a notion of Hahn product of compatible quasi-ordered groups and generalize Hahn's embedding theorem to quasi-ordered groups. We also develop a notion of quasi-order-minimality and establish a connection with C-minimality, thus answering a question of F. Delon. Finally, we use compatible quasi-ordered groups to give an example of a C-minimal group which is neither an ordered nor a valued group. [ABSTRACT FROM AUTHOR]

An important consequence of the Hahn–Banach theorem says that on any locally convex Hausdorff topological space X, there are sufficiently many continuous linear functionals to separate points of X. In the paper, we establish a "local" version of this theorem. The result is applied to study the uo-dual of a Banach lattice that was recently introduced in Gao et al. (Positivity 22(3):711–725, 2018). We also provide a simplified approach to the measure-free characterization of uniform integrability established in Kardaras (J Funct Anal 266:1913–1927, 2014). [ABSTRACT FROM AUTHOR]

The Sandwich theorem (König in Archiv der Mathematik 23:500-1972, 1972) yields the existence of a linear functional sandwiched in between a given superlinear functional and a given sublinear functional. By combining an argument implicit in Fuchssteiner and Maitland Wright (Q J Math 28:155-162, 1977) with a theorem of Pataraia (in: Presented at the 65th peripatetic seminar on sheaves and logic, Aarhus, Denmark, November, 1997), we show that such a linear functional obtains as a common fixed point of a certain family of mappings. [ABSTRACT FROM AUTHOR]

In this paper our main goal is showing that many of quantization results in functional analysis are rather algebraic. Following Esslamzadeh and Taleghani (Linear Algebra Appl 438:1372-1392, 2013), we call every subspace [resp. self-adjoint unital subspace] of a unital ∗-algebra, a quasi operator space [resp. quasi operator system]. Local operator systems can be realized as quasi operator spaces. Arveson's extension theorem asserts that B(H) is an injective object in the category of operator systems. We show that Arveson's theorem remains valid in the much larger category of quasi operator systems. This shows that Arveson's theorem as a non commutative extension of Hahn-Banach theorem, is of purely algebraic nature. Moreover we prove an algebraic extension of Ruan's theorem which gives a charcterization of bounded quasi operator spaces. Then we identify the largest and the smallest of such quasi quantizations of a seminormed space X, which we call QMAX(X) and QMIN(X). [ABSTRACT FROM AUTHOR]

This paper starts by recalling the author’s results on polynomial approximation over a Cartesian product A of closed unbounded intervals and its applications to solving Markov moment problems. Under natural assumptions, the existence and uniqueness of the solution are deduced. The characterization of the existence of the solution is formulated by two inequalities, one of which involves only quadratic forms. This is the first aim of this work. Characterizing the positivity of a bounded linear operator only by means of quadratic forms is the second aim. From the latter point of view, one solves completely the difficulty arising from the fact that there exist nonnegative polynomials on ℝn, n≥2, which are not sums of squares.

From the change of variable formula on the Wiener space, we calculate various integral transforms for functionals on the Wiener space. However, not all functionals can be obtained by using this formula. In the process of calculating the integral transform introduced by Lee, this formula is also used, but it is also not possible to calculate for all the functionals. In this paper, we define a generalized integral transform. We then introduce a new method to evaluate the generalized integral transform for functionals using series expressions. Our method can be used to evaluate various functionals that cannot be calculated by conventional methods.

The article discusses operators in Banach spaces for Jordan matrix, focusing on stabilization of natural numbers, functions and infinity. Topics include semigroup of mathematical objects, application of topological vector spaces and infinite dimension of Hahn-Banach theorem for continuous linear application.

A special class of separated normed cones, which includes convex cones in normed spaces and in spaces with an asymmetric norm, is distinguished on the basis of the functional separability of elements. It is shown that, generally, separated normed cones admit no linear injective isometric embedding in any normed space. An analog of the Banach-Mazur theorem on a sublinear injective embedding of a separated normed cone in the cone of real nonnegative continuous functions on the interval [0; 1] with the ordinary sup-norm is obtained. This result is used to prove the existence of a countable total set of bounded linear functionals for a special class of separated normed cones. [ABSTRACT FROM AUTHOR]

The paper examines different kinds of p-convexity of a function g which are sufficient for the existence of a linear functional such that in Theorem 1.13 of Simons in his monograph ‘From Hahn-Banach to monotonicity’, published in Springer lecture notes (2008). We replace sublinearity of p with convexity, the field with Dedekind vector lattice and present -convexity which is also necessary. In Theorem 4.7 we also generalize a result of MM. Neumann from 1991 published in Czech. Mathem. Journal Vol 41 on the Mazur-Orlicz theorem. [ABSTRACT FROM AUTHOR]

In set theory, an ideal is a collection of sets that are considered to be small or negligible, such that every subset of an element of the ideal must also be in the ideal and the union of any two elements of the ideal must also be in the ideal. A fuzzy set is a class of objects with grades of membership in the interval [0, 1]. It is used to mathematically represent uncertainty and provide a formal tool to deal with imprecisions present in many problems. We use ideals to def ine fuzzy on ideal sets, which can be seen as a generalization of the fuzzy sets. We establish some of its basic properties, and we state and prove a Hahn-Banach Theorem with the fuzzy on ideal sets, which can be seen as a generalization of a fuzzy Hahn-Banach Theorem, which in turn, is a fuzzif ied generalization of an analytic form of the classical Hahn-Banach Theorem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the discrete Hahn polynomials {Hn} and investigate their application for numerical solutions of the time fractional variable order mobile-immobile advection-dispersion model which is advantageous for modeling dynamical systems. This paper presented the operational matrix of derivative of discrete Hahn polynomials. The main advantage of approximating a continuous function by Hahn polynomials is that they have a spectral accuracy in the interval [0, N]. Furthermore, for computing the coefficients of the expansion u(x)=∑n=0∞cnHn(x), we have to only compute a summation and the calculation of coefficients is exact. Also an upper bound for the error of the presented method, with equidistant nodes, is investigated. Illustrative examples are provided to show the accuracy and efficiency of the presented method. Using a small number of Hahn polynomials, significant results are achieved which are compared to other methods. [ABSTRACT FROM AUTHOR]

The article discusses an ax-kochen-ershov theorem for monotone differential-henselian fields. It discusses Ax-Kochen-Ershov type results for differential-henselian monotone valued differential fields with many constants. It shows how to get rid of the condition with many constants. It keeps k as a differential field characteristic 0 with a single distinguished derivation along with an ordered abelian group Γ which gives rise to the Hahn field K = k((tΓ)).

In this paper we focus on developing bounded quasilinear interval-valued functions. We deal with the Hahn-Banach extension theorem for interval-valued functions. Finally, we show that there are enough bounded quasilinear functionals on the space of interval-valued functions to distinguish between its the regular elements. [ABSTRACT FROM AUTHOR]

We define convexity canonically in the setting of monoids. We show that many classical results from convex analysis hold for functions defined on such groups and semigroups, rather than only vector spaces. Some examples and counter-examples are also discussed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to generalize and unify in a single statement some results on the existence of a solution for an equilibrium problem, as well as to apply it to derive consequences in minimax theory and variational analysis. [ABSTRACT FROM AUTHOR]

In this paper, we propose an approach to the solution of the inverse problem for finding the initial function for the string vibration equation. The search for unknown initial function is reduced to the minimization problem of the functional, constructed with the help of the additional information. A formula is obtained for the gradient of the functional and a necessary and sufficient condition for optimality is derived. [ABSTRACT FROM AUTHOR]

In set theory without the axiom of choice [ABSTRACT FROM AUTHOR]

Recent developments have clarified that some tools of Convex Geometry are closely related to separation theorems obtained in the field of Functional Inequalities. This phenomenon has motivated the investigation of convex structures induced by Chebyshev systems. The present note characterizes such a possible structure, completely describing its combinatorial invariants. [ABSTRACT FROM AUTHOR]