A new form of the Hahn-Banach Theorem

The Hahn-Banach Theorem: a proof of the equivalence between the analitic and geometric versions

We present here a simple and direct proof of the classic geometric version of Hahn-Banach Theorem from its analitic version, in the real case. The reciprocal implication, and the direct proofs of both versions, are already well kown, but they are also summarized. For the complex case, in both versions the Hahn-Banach Theorem is deduced from the real case, as it is well known.

Hahn-Banach theorems for $κ$-normed spaces

For a new class of topological vector spaces, namely $κ$-normed spaces, and associated quasisemilinear topological preordered space is defined and investigated. This structure arise naturally from the consideration of a $κ$-norm, that is a distance function between a point and a $G_δ$-subset. For it, analogs of the Hahn-Banach theorem are proved.

A Local Hahn-Banach Theorem and Its Applications

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 [3]. We also provide a simplified approach to the measure-free characterization of uniform integrability established in [8].

Some Applications of the Hahn-Banach Separation Theorem

We show that a single special separation theorem (namely, a consequence of the geometric form of the Hahn-Banach theorem) can be used to prove Farkas type theorems, existence theorems for numerical quadrature with positive coefficients, and detailed characterizations of best approximations from certain important cones in Hilbert space.

Revisiting the Hahn-Banach Theorem and Nonlinear Infinite Programming

[REVISED VERSION] The aim of this paper is to state a sharp version of the König supremum theorem, an equivalent reformulation of the Hahn--Banach theorem. We apply it to derive statements of the Lagrange multipliers, Karush-Kuhn-Tucker and Fritz John type, for nonlinear infinite programs. We also show that a weak concept of convexity coming from minimax theory, infsup-convexity, is the adequate one for this kind of results.

Uniform Boundedness Principle and Hahn-Banach Theorem for b-linear functional related to linear 2-normed space

In this paper, we will see that the Cartesian product of two 2-Banach spaces is also 2-Banach space and discuss some properties of closed linear operator in linear 2-normed space. We also describe the concept of different types of continuity of b-linear functional and derive the Uniform Boundedness Principle and Hahn-Banach extension theorem for b-linear functionals in the case of linear 2-normed spaces. We also introduce the notion of weak*convergence for the sequence of bounded b-linear functionals relative to linear 2-normed space.

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.

关于Banach空间中的Hahn-Banach光滑性及相关性质

On Hahn-Banach smoothness and related properties in Banach spaces

本文研究Banach空间中Hahn-Banach光滑性的三类变体，刻画了相关性质，推导了强弱性质等价的充分条件，证明其可分决定性，并刻画了经典序列空间中对应子空间的特征。

A geometric form of the Hahn-Banach extension theorem for L0 linear functions and the Goldstine-Weston theorem in random normed modules

In this paper, we present a geometric form of the Hahn-Banach extension theorem for $L^{0}-$linear functions and prove that the geometric form is equivalent to the analytic form of the Hahn-Banach extension theorem. Further, we use the geometric form to give a new 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 $(ε,λ)-$topology and the locally $L^{0}-$convex topology, and also provide a counterexample s…

非线性广义函数空间中的 Hahn-Banach 定理

The Hahn-Banach theorem in spaces of nonlinear generalized functions

本文在 Colombeau 空间框架内建立了适用版本的 Hahn-Banach 定理，通过处理映射按 $\varepsilon$ 逐项定义的情形简化了线性泛函的延拓，并以此证明了该空间中凸集的分离。

Uniqueness of Hahn--Banach extensions and some of its variants

In this study, we analyze the various strengthening and weakening of the uniqueness of the Hahn--Banach extension. In addition, we consider the case in which $Y$ is an ideal of $X$. In this context, we study the property-$(U)/ (SU)/ (HB)$ and property-$(k-U)$ for a subspace $Y$ of a Banach space $X$. We obtain various new characterizations of these properties. We discuss various examples in the classical Banach spaces, where the aforementioned properties are satisfied and where they fail. It is observed that a hyperplane in $c_0$ has property-$(HB)$ if and only if it is an $M$-summand. Conside…

关于 $L_1$-预对偶空间的 Hahn-Banach 平滑性及相关 $w^*-w$ 点连续性问题

On Hahn-Banach smoothness of $L_1$-preduals and related $w^*-w$ point of continuity of unit balls of dual spaces

本文研究了 Banach 空间 Hahn-Banach 平滑性及其几何特性，探讨了 $L_1$-预对偶空间和仿射连续函数空间中线性泛函唯一norm保持扩展的条件，揭示了 Hahn-Banach 平滑性在特定空间中的结构特征。

多面体范数与光滑 Hahn-Banach 延拓

Polyhedral norms and smooth Hahn-Banach extension

本文在多面体范数下，给出了子空间上的光滑泛函存在保范光滑延拓到全空间的充要条件，即该光滑泛函的唯一绝对范数达到点是子空间单位球和全空间单位球的极点，并证明了该结果在非多面体范数下不成立。

Hahn-Banach for metric functionals and horofunctions

It is observed that a natural analog of the Hahn-Banach theorem is valid for metric functionals but fails for horofunctions. Several statements of the existence of invariant metric functionals for individual isometries and 1-Lipschitz maps are proved. Various other definitions, examples and facts are pointed out related to this topic. In particular it is shown that the metric (horofunction) boundary of every infinite Cayley graphs contains at least two points.

非对称赋范空间上线性算子的Hahn-Banach型延拓结果

Hahn-Banach type extension results for linear operators on asymmetric normed spaces

本文针对非对称赋范空间上线性算子的延拓问题，将Nachbin的经典思路适配到非对称情形，给出了混合二元交性质与非对称赋范空间单射性等价这一已有结论的直接证明，丰富了非对称赋范空间下的Hahn-Banach型理论。

Decompositions, approximate structure, transference, and the Hahn-Banach theorem

This paper is partly a survey of certain kinds of results and proofs in additive combinatorics, and partly a discussion of how useful the finite-dimensional Hahn-Banach theorem can be. The most interesting single result is probably a simpler proof of a key step in the proof of the Green-Tao theorem, but several other applications of the method are given. A similarly simplified proof of the Green-Tao transference principle was obtained independently (and expressed in a rather different language) by Reingold, Trevisan, Tulsiani and Vadhan.

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 Konig's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multi-valued function Sep and a natural notion of reducibility for multi-valued functions, we obtain a computational counterpart of the subsystem of second order arithmetic WKL_0. We study analogies and differences between WKL_0 and the class of Sep-computable mul…

实C$^*$-代数和实JB$^*$-三元组中Hahn--Banach扩张的唯一性与内理想

Uniqueness of Hahn--Banach extensions and inner ideals in real C$^*$-algebras and real JB$^*$-triples

本文研究实JB$^*$-三元组中Hahn--Banach光滑性与内理想的关系，证明闭子三元组光滑当且仅当遗传，并在一定条件下由光滑性推出内理想。

Haagerup问题的一个解与算子代数中的正Hahn-Banach分离定理

A solution to Haagerup's problem and positive Hahn-Banach separation theorems in operator algebras

该研究解决了Uffe Haagerup1975年提出的C*-代数对偶空间上正极定理问题，得到四类空间上的正Hahn-Banach分离定理，还简化了Dixmier正规权问题的原始解答。

Uniqueness of Hahn-Banach extension and related norm-$1$ projections in dual spaces

In this paper we study two properties viz. property-$U$ and property-$SU$ of a subspace $Y$ of a Banach space which correspond to the uniqueness of the Hahn-Banach extension of each linear functional in $Y^*$ and in addition to that this association forms a linear operator of norm-1 from $Y^*$ to $X^*$. It is proved that, under certain geometric assumptions on $X, Y, Z$ these properties are stable with respect to the injective tensor product; $Y$ has property-$U$ ($SU$) in $Z$ if and only if $X\otimes_\e^\vee Y$ has property-$U$ ($SU$) in $X\otimes_\e^\vee Z$. We prove that when $X^*$ has the …

Extreme flatness and Hahn-Banach type theorems for normed modules over c_0

Let A be a commutative normed algebra, K a class of normed A-modules. A normed A-module Z is called extremely flat with respect to K, if, for every isometric morphism of normed A-modules, belonging to K, the non-completed projective A-module tensor product of this morphism and the identity morphism on Z, is also isometric. In the present paper we take, in the capacity of A, the algebra c_0 of vanishing sequences and consider the class of the so-called homogeneous modules, over the latter algebra, denoted by H. The main theorem gives a full description of essential homogeneous modules over the …

哈恩-巴拿赫定理如何照亮经典热力学中的基本问题

How the Hahn-Banach Theorem Sheds Bright Light on Fundamental Questions in Classical Thermodynamics

本文展示哈恩-巴拿赫定理如何从热力学第二定律直接导出熵和热力学温度函数，并指出唯一性需要可逆过程覆盖所有状态。

保序唯一Hahn-Banach延拓

Order-preserving unique Hahn-Banach extensions

本文研究带单位元的实Banach格中子空间的序与等距结构，通过紧凸集及Choquet单纯形的相关条件，得出正泛函延拓唯一性对应的集合特性结论。

The order versions of the Hahn--Banach Theorem and envelopes. I. Homogeneous functions

We present here the General formulation of the problem of existence and construction of upper and lower envelope for an arbitrary function with values from the completion of the ordered set ${\rm S}$ for a certain class of functions with values in ${\rm S}$. The task is parsed only for the simplest case of model class of homogeneous functions. Consider only order-algebraic version without the involvement of the topology.

局部凸空间中Hahn-Banach延拓的唯一性

Uniqueness of Hahn-Banach extensions in locally convex spaces

本文研究局部凸空间中线性泛函的Hahn-Banach延拓唯一性，提出property-SNP并给出其多种刻画，建立其与Banach空间property-U的对应关系，还在逐点收敛拓扑下的连续函数空间上开展相关研究。

Hahn--Banach Theorem and Duality Theory on non-Archimedean Locally Convex Spaces

Let $k$ be a local field with valuation ring $O_k$ and residue field $\overline{k}$. We extend Hahn--Banach theorem for the class of seminormed $k$-vector spaces to several classes of locally convex spaces and subspaces over $k$, $O_k$, and $\overline{k}$. We establish analogues of Iwasawa-type duality for several classes of locally convex spaces over $k$, $O_k$, and $\overline{k}$.

Extendibility of bilinear forms on Banach sequence spaces

We study Hahn-Banach extensions of multilinear forms defined on Banach sequence spaces. We characterize $c_0$ in terms of extension of bilinear forms, and describe the Banach sequence spaces in which every bilinear form admits extensions to any superspace.

非平衡经典热力学中熵与热力学温度作为Hahn-Banach定理直接推论的研究：I. 存在性

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

该研究从热力学第二定律的Kelvin-Planck表述出发，借助Hahn-Banach定理直接证明非平衡态下局部比熵与热力学温度状态函数的存在性，无需限定平衡态条件，相关性质将在配套论文展开。

Order version of the Hahn-Banach theorem, and envelopes. II. Applications to the function theory

Chapter 1 deals with the problem of the existence of an upper/lower envelope from 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. Vector, ordinal and topological dual interpretations of the existence conditions of such envelope and the method of its construction are given. Chapter 2 presents applications to the existence of a nontrivial (pluri)subharmonic and/or (pluri) harmonic minorant for functions in domains from finite-dimensional real or complex spa…

非平衡经典热力学中作为 Hahn-Banach 定理直接推论的熵与热力学温度：II. 性质

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

本文研究由 Hahn-Banach 定理从 Kelvin-Planck 第二定律导出的非平衡熵与热力学温度，重点分析温度对冷热程度的反映、熵与温度的本质唯一性，以及温度计的定义和存在所带来的性质。

关于完全光滑重赋范的一个注记

A remark on totally smooth renormings

本文指出 Oja 等人关于弱紧生成 Banach 空间完全光滑重赋范的结果在无空间限制的一般情形下仍成立，且形式更强，这得益于 M. Raja 的相关结论。

Representations of dual spaces

We give a nonlinear representation of the duals for a class of Banach spaces. This leads to classroom-friendly proofs of the classical representation theorems $H'=H$ and $(L^p)'=L^q$. Our proofs extend to a family of Orlicz spaces, and yield as an unexpected byproduct a version of the Helly-Hahn-Banach theorem.

Bicomplex Linear Operators on Bicomplex Hilbert Spaces and Littlewood's Subordination Theorem

In this paper we study some basic properties of bicomplex linear operators on bicomplex Hilbert spaces. Further we discuss some applications of Hahn-Banach theorem on bicomplex Banach modules. We also introduce and discuss some bicomplex holomorphic function spaces and prove Littlewood's Subordination principle for bicomplex Hardy space.

Noncommutative Vitali-Hahn-Saks Theorem holds precisely for finite $W^\ast$-algebras

It is shown that the bona fide generalization of the Vitali-Hahn-Saks Theorem to von Neumann algebras is possible if, and only if, the algebra is finite. This settles the problem on the noncommutative Vitali-Hahn-Saks Theorem completely and provides new means of characterizing finite von Neumann algebras.

The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra

This article investigates matrix convex sets and introduces their tracial analogs which we call contractively tracial convex sets. In both contexts completely positive (cp) maps play a central role: unital cp maps in the case of matrix convex sets and trace preserving cp (CPTP) maps in the case of contractively tracial convex sets. CPTP maps, also known as quantum channels, are fundamental objects in quantum information theory. Free convexity is intimately connected with Linear Matrix Inequalities (LMIs) L(x) = A_0 + A_1 x_1 + ... + A_g x_g > 0 and their matrix convex solution sets { X : L(X) …

没有‘选择’的生活

Life without "Choice"

本文基于Hahn-Banach定理提出一个不依赖选择公理的非测集存在的证明，该定理严格弱于选择公理。

A new and simple proof of Schauder's theorem

Schauder's theorem asserts that a bounded linear operator between Banach spaces is compact if ad only if its adjoint is. We give a new proof of this result, which is both short and completely elementary in the sense that it does not depend on anything beyond basic functional analysis, i.e., the Hahn--Banach theorem and some of its consequences; in particular, we avoid the Arzela--Ascoli theorem (and any kind of related diagonal argument).

A Hahn-Jordan decomposition and Riesz-Frechet representation theorem in Riesz spaces

We give a Hahn-Jordan decomposition in Riesz spaces which generalizes that of [{{\sc B. A. Watson}, {An Andô-Douglas type theorem in Riesz spaces with a conditional expectation,} {\em Positivity,} {\bf 13} (2009), 543 - 558}] and a Riesz-Frechet representation theorem for the $T$-strong dual, where $T$ is a Riesz space conditional expectation operator. The result of Watson was formulated specifically to assist in the proof of the existence of Riesz space conditional expectation operators with given range space, i.e., a result of Andô-Douglas type. This was needed in the study of Markov process…

On Zippin's Embedding Theorem of Banach spaces into Banach spaces with bases

We present a new proof of Zippin's Embedding Theorem, that every separable reflexive Banach space embeds into one with shrinking and boundedly complete basis, and every Banach space with a separable dual embeds into one with a shrinking basis. This new proof leads to improved versions of other embedding results.

Weak convergence theorems for symmetric generalized hybrid mappings in uniformly convex Banach spaces

In this paper, we prove some theorems related to properties of generalized symmetric hybrid mappings in Banach spaces. Using Banach limits, we prove a fixed point theorem for symmetric generalized hybrid mappings in Banach spaces. Moreover, we prove some weak convergence theorems for such mappings by using Ishikawa iteration method in a uniformly convex Banach space.

Lyapunov theorem for q-concave Banach spaces

Generalization of Lyapunov convexity theorem is proved for vector measure with values in Banach spaces with unconditional bases, which are q-concave for some $q<\infty.$

Linear functionals on idempotent spaces: an algebraic approach

In this paper, we present an algebraic approach to idempotent functional analysis, which is an abstract version of idempotent analysis. The basic concepts and results are expressed in purely algebraic terms. We consider idempotent versions of certain basic results of linear functional analysis, including the theorem on the general form of a linear functional and the Hahn-Banach and Riesz-Fischer theorems.

Banach-Stone型定理在一致连续和Lipschitz连续伪度量空间上的应用

Banach-Stone type theorems on uniformly continuous and lipschitz continuous pseudometrics

本文研究一致连续和Lipschitz连续伪度量空间中的Banach-Stone型定理，探讨其核心方法与主要贡献。

The Bolzano-Poincaré-Miranda theorem in infinite dimensional Banach spaces

We study the existence of zeroes of mappings defined in Banach spaces. We obtain, in particular, an extension of the well-known Bolzano-Poincaré-Miranda theorem to infinite dimensional Banach spaces. We also establish a result regarding the existence of periodic solutions to differential equations posed in an arbitrary Banach space.

Best proximity points and weak convergence theorems for hybrid mappings in Banach spaces

The paper gives some results on best proximity and fixed point for a class of generalized hybrid cyclic self-mappings in Banach spaces.

A bicommutant theorem for dual Banach algebras

A dual Banach algebra is a Banach algebra which is a dual space, with the multiplication being separately weak$^*$-continuous. We show that given a unital dual Banach algebra $\mc A$, we can find a reflexive Banach space $E$, and an isometric, weak$^*$-weak$^*$-continuous homomorphism $π:\mc A\to\mc B(E)$ such that $π(\mc A)$ equals its own bicommutant.

Group invariant operators and some applications on norm-attaining theory

In this paper, we study geometric properties of the set of group invariant continuous linear operators between Banach spaces. In particular, we present group invariant versions of the Hahn-Banach separation theorems and elementary properties of the invariant operators. This allows us to contextualize our main applications in the theory of norm-attaining operators; we establish group invariant versions of the properties $α$ of Schachermayer and $β$ of Lindenstrauss, and present relevant results from this theory in this (much wider) setting. In particular, we generalize Bourgain's result, which …

Ergodic theorems for $L_1$--$L_\infty$ contractions in Banach--Kantorovich $L_p$-lattices

We present versions of ergodic theorems for $L_1$--$L_\infty$ contractions in Banach--Kantorovich $L_p$-lattices associated with the Maharam measure taking values in the algebra of measurable functions.

A Strong Convergence Theorem for finite families of Bregman Demimetric Mappings in a Banach Space under a New Shrinking projection Method

A Strong Convergence Theorem for finite families of Bregman Demimetric Mappings in a Banach Space under a New Shrinking projection Method

A Banach Algebraic Version Of The Fundamental Theorem of Algebra

For a monic polynomial p(z) with coefficients in a unital complex Banach algebra, we prove that there exist a complex number z such that p(z)is not invertible

Coincidence points of Mappings in Banach Spaces

In this article we prove an existence theorem for coincidence points of mappings in Banach spaces. This theorem generalizes the Kantorovich fixed point theorem.

A strong open mapping theorem for surjections from cones onto Banach spaces

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of Andô's Theorem for an ordered Banach spa…

Zippin's embedding theorem and amalgamations of classes of Banach spaces

It was proved by Dodos and Ferenczi that the classes of Banach spaces with a separable dual and of separable reflexive Banach spaces are strongly bounded. In this note, we provide an isometric version of this result.

The Besicovitch-Federer projection theorem is false in every infinite dimensional Banach space

We construct a purely unrectifiable set of finite $\mathcal H^1$-measure in every infinite dimensional separable Banach space $X$ whose image under every $0\neq x^*\in X^*$ has positive Lebesgue measure. This demonstrates completely the failure of the Besicovitch-Federer projection theorem in infinite dimensional Banach spaces.