David J. Foulis, Sylvia Pulmannová
We give a detailed proof D. Handelman's theorem stating (in the context of an order unit normed space) that a monotone sigma-complete order unit normed space is a Banach space.
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Normed linear spaces which are isometric to order unit spaces
通过构造实赋范线性空间与序单位空间、基赋范空间的线性直和，本文刻画了可等距同构于序单位空间的赋范空间，并证明 ℓ₁ 与 ℓ∞ 均可而 ℓ₂ 不可。
We suggest a concept of generalized `angles' in arbitrary real normed vector spaces. We give for each real number a definition of an `angle' by means of the shape of the unit ball. They all yield the well known Euclidean angle in the special case of real inner product spaces. With these different angles we achieve a classification of normed spaces, and we obtain a characterization of inner product spaces. Finally we consider this construction also for a generalization of normed spaces, i.e. for spaces which may have a non-convex unit ball.
Polyhedral normed spaces: the structural theorem and locally finite tilings
本文证明具有($\Delta$)性质的多面体赋范空间的单位球面被代数真面覆盖，并推广到(VI)-多面体情形，进而刻画了局部有限铺砌的存在性，推广了Fonf的结果。
We characterize $M$-ideals in order smooth $\infty$-normed spaces by extending the notion of split faces of the state space to those of the quasi-state space. We also characterize approximate order unit spaces as those order smooth $\infty$-normed spaces $V$ that are $M$-ideals in $\tilde{V}.$ Here $\tilde{V}$ is the order unit space obtained by adjoining an order unit to $V.$ To prove these results, we develop an order theoretic version of the "Alfsen-Efffros' cone decomposition theorem" for order smooth $1$-normed spaces. (As a quick application of this result, we sharpen a result on the ext…
In this note we generalize the well-known Wigner's unitary-anti\-unitary theorem. For $X$ and $Y$ smooth normed spaces and $f:X\to Y$ a surjective mapping such that $|[f(x),f(y)]|=|[x,y]|$, $x,y\in X$, where $[\cdot,\cdot]$ is the unique semi-inner product, we show that $f$ is phase equivalent to either a linear or an anti-linear surjective isometry. When $X$ and $Y$ are smooth real normed spaces and $Y$ strictly convex, we show that Wigner's theorem is equivalent to $\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\{\|x+y\|,\|x-y\|\}$, $x,y\in X$.
Rigidity of unit spheres in finite-dimensional real normed spaces
本文证明有限维实赋范空间中单位球面等距蕴含线性等距，通过Plücker收缩体和零拉格朗日恒等式回答Kadets-Martín问题。
On strong orthogonality and strictly convex normed linear spaces
本文在赋范线性空间中引入 Birkhoff-James 意义下的强正交集概念，给出了单位球面上元素成为暴露点的充要条件，并证明了空间严格凸性与有界线性算子仅在特定点达到范数的等价关系。
Let $S$ denote the unit sphere of a real normed space. We show that the intrinsic metric on $S$ is strongly equivalent to the induced metric on $S$. Specifically, for all $x,y\in S$, \[ \|x-y\|\leq d(x,y)\leq\sqrt{2}π\|x-y\|, \] where $d$ denotes the intrinsic metric on $S$.
Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P:Y\to X$ such that $P(B_Y)\subset A$. The main results of the paper: {\bf (1)} Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. {\bf (2)} If a finite dimensional normed linear space has a minima…
We generalise the formula expressing the matrix trace of a given square matrix as the integral of the numerical values of $A$ over the Euclidean sphere to the unit spheres of finite-dimensional normed spaces that have a 1-symmetric basis. Our result is new even in the case of $\ell_p$-norms in $\mathbb{R}^N$ for $p\neq 2$.
In this paper, we define and study the space of all the functions of bounded variation $f:[x,y]\to \mathbb{Y}$ denoted by $\mathcal{BV}[x,y],$ where $[x,y]$ is an ordered interval and $\mathbb{Y}$ is an absolute order unit space having vector lattice structure. By default, under the order structure of $\mathbb{Y},$ the space $\mathcal{BV}[x,y]$ forms a nearer absolute order unit space structure and in some cases it turns out to be an absolute order unit space (in fact, a unital $AM$-space). By help of variation function, we also define a different kind of order structure on the space $\mathcal…
The quotient shape types of normed vectorial spaces(over the same field) with respect to Banach spaces reduce to those of Banach spaces. The finite quotient shape type of normed spaces is an invariant of the (algebraic) dimension, but not conversely. The converse holds for separable normed spaces as well as for the bidual-like spaces (isomorphic to their second dual spaces). As a consequence, the Hilbert space $l_2$, or even its (countably dimensional, unitary) direct sum subspace may represent the unique quotient shape type of all $2^\aleph_0$-dimensional normed spaces. An application yields …
Using a technique of adjoining an order unit to a normed linear space, we have characterized strictly convex spaces among normed linear spaces and Hilbert spaces among strictly convex Banach spaces respectively. This leads to a generalization of spin factors and provides a new class of absolute order unit spaces.
On rectangular constant in normed linear spaces
本文研究了赋范线性空间中矩形常数的性质，证明了其取值为3及达到上界的充要条件是单位球面包含长度为2的线段，并给出了空间为内积空间的充要条件。
Angular equivalence is introduced and shown to be an equivalence relation among the norms on a fixed real vector space. It is a finer notion than the usual (topological) notion of norm equivalence. Angularly equivalent norms share certain geometric properties: A norm that is angularly equivalent to a uniformly convex norm is itself uniformly convex. The same is true for strict convexity. Extreme points of the unit balls of angularly equivalent norms occur on the same rays, and if one unit ball is a polyhedron so is the other. Among norms arising from inner products, two norms are angularly equ…
The paper is concerned with compact bilinear operators on asymmetric normed spaces. The study of multilinear operators on asymmetric normed spaces was initiated by Latreche and Dahia, Colloq. Math. (2020). We go further in this direction and prove a Schauder type theorem on the compactness of the adjoint of a compact bilinear operator and study the ideal properties of spaces of compact bilinear operators. These extend some results of Ramanujan and Schock, Linear and Multilinear Algebra (1985), and Ruch, ibid. (1989), on compact bilinear operators on Banach spaces. On the space of bilinear form…
In this paper we prove in certain n-dimensional normed spaces X the existence of full-dimensional equilateral simplices of large size inscribed to the unit ball B. This extends the construction of Makeev [Mak] in dimension 4 and we also compute an example of a space in which the idea cannot be applied.
The space Weak L^1 consists of all measurable functions on [0,1] such that q(f) = sup_{c>0} c λ{t : |f(t)| > c} is finite, where λdenotes Lebesgue measure. Let ρbe the gauge functional of the unit ball {f : q(f) \leq 1} of the quasi- norm q, and let N be the null space of ρ. The normed envelope of Weak L^1, which we denote by W, is the space (Weak L^1/N, ρ). The Banach envelope of Weak L^1, \overline{W}, is the completion of W. We show that \overline{W} is isometrically lattice isomorphic to a sublattice of W. It is also shown that all rearrangement invariant Banach function spaces are isometr…
We study the sizes of delta-additive sets of unit vectors in a d-dimensional normed space: the sum of any two vectors has norm at most delta. One-additive sets originate in finding upper bounds of vertex degrees of Steiner Minimum Trees in finite dimensional smooth normed spaces (Z. Füredi, J. C. Lagarias, F. Morgan, 1991). We show that the maximum size of a delta-additive set over all normed spaces of dimension d grows exponentially in d for fixed delta>2/3, stays bounded for delta<2/3, and grows linearly at the threshold delta=2/3. Furthermore, the maximum size of a 2/3-additive set in d-dim…
This paper considers weak supercyclicity for bounded linear operators on a normed space. On the one hand, weak supercyclicity is investigated for classes of Hilbert-space operators: (i) self-adjoint operators are not weakly supercyclic, (ii) diagonalizable operators are not weakly l-sequentially supercyclic, and (iii) weak l-sequential supercyclicity is preserved between a unitary operator and its adjoint. On the other hand, weak supercyclicity is investigated for classes of normed-space operators: (iv) the point spectrum of the normed-space adjoint of a power bounded supercyclic operator is e…
We study some generalized metric properties of weak topologies when restricted to the unit sphere of some equivalent norm on a Banach space, and their relationships with other geometrical properties of norms. In case of dual Banach space $X^*$, we prove that there exists a dual norm such that its unit sphere is a Moore space for the weak$^*$-topology (has a G$_δ$-diagonal for the weak$^*$-topology, respectively) if, and only if, $X^*$ admits an equivalent weak$^*$-LUR dual norm (rotund dual norm, respectively).
In this paper we introduce the notion of a quasi-modular and we prove that the respective Minkowski functional of the unit quasi-modular ball becomes a quasi-norm. In this way, we refer to and complete the well-known theory related to the notions of a modular and a convex modular that lead to the F-norm and to the norm, respectively. We use the obtained results to consider basic properties of quasi-normed Calderón-Lozanovski\uı spaces $E_φ$, where the lower Matuszewska-Orlicz index $α_φ$ plays the key role. We also give a number of theorems concerning different copies of $l^{\infty}$ in the sp…
The uniform norm on a uniform normed Q-algebra is the only uniform Q-algebra norm on it. The uniform norm on a regular uniform normed Q-algebra with unit is the only uniform norm on it. Let A be a uniform topological algebra whose spectrum M (A) is equicontinuous, then A is a uniform normed algebra. Let A be a regular semisimple commutative Banach algebra, then every algebra norm on A is a Q-algebra norm on A.
Truncated Riesz spaces was first introduced by Fremlin in the context of real-valued functions. An appropriate axiomatization of the concept was given by Ball. Keeping only the first Ball's Axiom (among three) as a definition of truncated Riesz spaces, the first named author and El Adeb proved that if $E$ is truncated Riesz space then $E\oplus\mathbb{R}$ can be equipped with a non-standard structure of Riesz space such that $E$ becomes a Riesz subspace of $E\oplus\mathbb{R}$ and the truncation of $E$ is provided by meet with $1$. In the present paper, we assume that the truncated Riesz space $…
On isosceles orthogonality and some geometric constants in a normed space
该研究探究赋范空间中James常数与等腰正交性的关联，证明了James常数取极值时的正交性质，给出二维多面体Banach空间中该常数的极点取法，为其高效计算提供了方法，并拓展研究了相关几何常数问题。
Projections in an order unit space and orthogonality
本文在序单位空间中定义序投影，以几何正交性刻画其特征，并刻画由赋范线性空间添加序单位所得空间中的序投影。
We prove that, if the closed unit ball of a normed space $X$ has sufficiently many extreme points, then every mapping $Φ$ from $X$ into itself with the following property is affine: For any pair of points in $X$, there exists a (not necessarily linear) surjective isometry on $X$ that coincides with $Φ$ at the two points. We also consider surjectivity of such a mapping in some special cases including C$^*$-algebras.
The \emph{relative projection constant} $λ(Y, X)$ of normed spaces $Y \subset X$ is defined as $λ(Y, X) = \inf \{ ||P|| : P \in \mathcal{P}(X, Y) \}$, where $\mathcal{P}(X, Y)$ denotes the set of all continuous projections from $X$ onto $Y$. By the well-known result of Bohnenblust for every $n$-dimensional normed space $X$ and its subspace $Y$ of codimension $1$ the inequality $λ(Y, X) \leq 2 - \frac{2}{n}$ holds. The main goal of the paper is to study the equality case in the theorem of Bohnenblust. We establish an equivalent condition for the equality $λ(Y, X) = 2 - \frac{2}{n}$ and present …
We introduce the notion centre of a convex set and study the space of continuous affine functions on a compact convex set with a centre. We show that these spaces are precisely the dual of a base normed space in which the underlying base has a (unique) centre. We also characterize the corresponding base norm space. We obtain a condition on a compact, balanced, convex subset of a locally convex space so that the corresponding space of continuous affine functions on the convex set is an absolute order unit space. Similarly, we characterize a condition on the base with a centre of a base normed s…
Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P:Y\to X$ such that $P(B_Y)\subset A$. Each finite dimensional normed space has a minimal-volume sufficient enlargement which is a parallelepiped, some spaces have "exotic" minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having "exotic" …
It is well known that every (real or complex) normed linear space $L$ is isometrically embeddable into $C(X)$ for some compact Hausdorff space $X$. Here $X$ is the closed unit ball of $L^*$ (the set of all continuous scalar-valued linear mappings on $L$) endowed with the weak$^*$ topology, which is compact by the Banach-Alaoglu theorem. We prove that the compact Hausdorff space $X$ can indeed be chosen to be the Stone-Cech compactification of $L^*\setminus\{0\}$, where $L^*\setminus\{0\}$ is endowed with the supremum norm topology.
Let $x$ and $y$ be two unit vectors in a normed plane $\mathbb{R}^2$. We say that $x$ is Birkhoff orthogonal to $y$ if the line through $x$ in the direction $y$ supports the unit disc. A B-measure (Fankhänel 2011) is an angular measure $μ$ on the unit circle for which $μ(C)=π/2$ whenever $C$ is a shorter arc of the unit circle connecting two Birkhoff orthogonal points. We present a characterization of the normed planes that admit a B-measure.
Lower Bounds for the Shadiness Constant of Finite-Dimensional Normed Spaces
本文利用优化技术为有限维赋范空间的阴暗常数提供了可证明的下界，构造了阴暗常数至少为1.01的3维赋范空间，并证实了Bosznay和Garay关于不存在具有10个或更少顶点单位球的阴暗范数的猜想。
Definition. A symmetric with respect to 0 bounded closed convex set A in a finite dimensional normed space X is called a sufficient enlargement for X (or of B(X)) if for arbitrary isometric embedding of X into a Banach space Y there exists a projection P:Y\to X such that P(B(Y)) is a subset of A (by B(X) we denote the unit ball). The notion of sufficient enlargement is implicit in the paper: B.Grunbaum, Projection constants, Trans. Amer. Math. Soc. 95 (1960) 451--465. It was explicilty introduced by the author in: M.I.Ostrovskii, Generalization of projection constants: sufficient enlargements,…
The Voronoi diagram is a geometric object which is widely used in many areas. Recently it has been shown that under mild conditions Voronoi diagrams have a certain continuity property: small perturbations of the sites yield small perturbations in the shapes of the corresponding Voronoi cells. However, this result is based on the assumption that the ambient normed space is uniformly convex. Unfortunately, simple counterexamples show that if uniform convexity is removed, then instability can occur. Since Voronoi diagrams in normed spaces which are not uniformly convex do appear in theory and pra…
We give necessary and sufficient conditions for a nonexpansive map on a finite dimensional normed space to have a nonempty, bounded set of fixed points. Among other results we show that if $f : V \rightarrow V$ is a nonexpansive map on a finite dimensional normed space $V$, then the fixed point set of $f$ is nonempty and bounded if and only if there exist $w_1, \ldots , w_m$ in $V$ such that $\{f(w_i) - w_i : i = 1, \ldots, m \}$ illuminates the unit ball. This yields a numerical procedure for detecting fixed points of nonexpansive maps on finite dimensional spaces. We also discuss application…
We prove that every unital C*-algebra $A$ has the Mazur--Ulam property. Namely, every surjective isometry from the unit sphere $S_A$ of $A$ onto the unit sphere $S_Y$ of another normed space $Y$ extends to a real linear map. This extends the result of A. M. Peralta and F. J. Fernandez-Polo who have proved the same under the additional assumption that both $A$ and $Y$ are von Neumann algebras. In the course of the proof, we strengthen Mankiewicz's theorem and prove that every surjective isometry from a closed unit ball with enough extreme points onto an arbitrary convex subset of a normed space…
Fitting Gaussian vectors in normed spaces
本文研究一般赋范空间中的高斯向量拟合问题，重点分析ℓ_∞^d空间，证明cd²/log d个点可高概率拟合，发现二维ℓ_2²与ℓ_∞²空间具有极端可拟合性。
In this paper the weak topology on a normed space is studied from the viewpoint of infinite-dimensional topology. Besides the weak topology on a normed space $X$ (coinciding with the topology of uniform convergence on finite subsets of the dual space $X^*$), we consider the topology $c$ of uniform convergence on compact subsets of $X^*$. It is known that this topology coincides with the weak topology on bounded subsets of $X$, but unlike to the latter has much better topological properties (e.g., is stratifiable). We prove that for normed spaces $X,Y$ with separable duals the spaces $(X,weak)$…
Some results on the Dunkl-Williams constant
本文通过推导公式计算了实赋范线性空间中Dunkl-Williams常数DW(X)及与Birkhoff正交相关的常数DW_B(X)，并证明了ℓ₂-ℓ₁空间和十二面体单位球空间的DW(X)值分别为2√2和8(2−√3)。
Characterization of inner product spaces
本文通过最佳余逼近与最佳逼近的重合性质刻画了维数大于2的实内积空间，并提出了关于实光滑赋范空间中强正交Hamel基的猜想。
$2$-strong uniqueness of a best approximation and of minimal projections in complex polytope norms and their duals
本文研究了复多面体范数及其对偶空间中最佳逼近的$2$-强唯一性，证明两类空间不相交，并在特定条件下将该性质推广至极小投影，得出三维复空间二维子空间上范数大于$1$的极小投影是$2$-强唯一的。
The main aim of this paper is to prove a Bishop-Phelps-Bollobás type theorem on the unital uniform algebra A_{w^*u}(B_{X^*}) consisting of all w^*-uniformly continuous functions on the closed unit ball B_{X^*} which are holomorphic on the interior of B_{X^*}. We show that this result holds for A_{w^*u}(B_{X^*}) if X^* is uniformly convex or X^* is the uniformly complex convex dual space of an order continuous absolute normed space. The vector-valued case is also studied.
Higher traces as boundary averages on finite-dimensional normed spaces
研究\(N\)维实赋范空间\(X\)中\(V = \Lambda^kX\)单位球面上满足特定等式的概率测度\(\eta\)，通过刻画得出锥概率测度等满足条件，还得到相关公式及结论，如锥测度是普遍各向同性成员等。
We describe absolutely ordered $p$-normed spaces, for $1 \le p \le \infty$ which presents a model for "non-commutative" vector lattices and includes order theoretic orthogonality. To demonstrate its relevance, we introduce the notion of {\it absolute compatibility} among positive elements in absolute order unit spaces and relate it to symmetrized product in the case of a C$^{\ast}$-algebra. In the latter case, whenever one of the elements is a projection, the elements are absolutely compatible if and only if they commute. We develop an order theoretic prototype of the results. For this purpose…
We say that a map $T: S_X\rightarrow S_Y$ between the unit spheres of two real normed-spaces $X$ and $Y$ is a phase-isometry if it satisfies \begin{eqnarray*} \{\|T(x)+T(y)\|, \|T(x)-T(y)\|\}=\{\|x+y\|, \|x-y\|\} \end{eqnarray*} for all $x,y\in S_X$. In the present paper, we show that there is a phase function $\varepsilon:S_X\rightarrow \{-1,1\}$ such that $\varepsilon \cdot T$ is an isometry which can be extended a linear isometry on the whole space $X$ whenever $T$ is surjective, $X=C(K)$ ($K$ is a compact Hausdorff space) and $Y$ is an arbitrary Banach space. Additionally, if $T$ is a phas…
Stable topology and stable compactness
本文针对随机泛函分析中的非紧问题，发展稳定集合论，提出稳定拓扑空间与稳定紧空间理论，推广紧性相关定理并建立多个随机版本的经典泛函分析定理。
Let $X$ be a real normed space with unit sphere S. We prove that $X$ is an inner product space if and only if there exists a real number $ρ=\sqrt{(1+\cos\frac{2kπ}{2m+1})/2}$, $(k=1,2,\ldots , m ;\:m=1,2,\ldots)$, such that every chord of $S$ that supports $ρS$ touches $ρS$ at its middle point. If this condition holds, then every point $u\in S$ is a vertex of a regular polygon that is inscribed in $S$ and circumscribed about $ρS$.
Let H(X) be the generalized Heisenberg group induced by a normed space X. We prove that X is a relatively minimal subgroup of H(X). We show that the group $G:=H(L_4[0,1])$ is reflexively representable but weakly continuous unitary representations of G in Hilbert spaces do not separate points of G. This answers a question of A. Shtern.
On the set of extreme points of the unit ball of a Hardy-Lorentz space
本文证明在满足特定条件的Lorentz空间中，模为1的函数是单位球极值点，并补充了de Leeuw-Rudin定理，指出H^1是唯一具有此性质的Hardy-Lorentz空间，同时给出非切向极限为常数的函数为极值点的充分条件。
One of the oldest and richest problems from continuous location science is the famous Fermat-Torricelli problem, asking for the unique point in Euclidean space that has minimal distance sum to n given (non-collinear) points. Many natural and interesting generalizations of this problem were investigated, e.g., by extending it to non-Euclidean spaces and modifying the used distance functions, or by generalizing the configuration of participating geometric objects. In the present paper, we extend the Fermat-Torricelli problem in a two-fold way: more general than for normed spaces, the unit balls …
Following a recent idea by Ball, we introduce the notion of strongly truncated Riesz space with a suitable spectrum. We prove that, under an extra Archimedean type condition, any strongly truncated Riesz space is isomorphic to a uniformly dense Riesz subspace of a $C_{0}\left( X\right) $-space. This turns out to be a direct generalization of the classical Kakutani Representation Theorem on Archimedean Riesz spaces with strong unit. Another representation theorem on normed Riesz spaces, due to Fremlin, will be obtained as a consequence of our main result.
An Exposition on Weak Stability of Operators
本文综述赋范空间（尤其是希尔伯特空间）上有界线性算子的弱稳定性研究，梳理六十年文献并指出研究集中于希尔伯特空间算子及酉算子弱稳定性的核心地位。
In an $n$-dimensional normed space every bounded set has a unique circumball if and only if every set of cardinality two has a unique circumball and if and only if the unit ball of the space is strictly convex. When the symmetry of the norm is dropped, i.e., when the centrally symmetric unit ball is replaced by an arbitrary convex body, then the above three conditions are no longer equivalent. We show for the latter case that every bounded set has a unique circumball if and only if every set of cardinality at most $n$ has a unique circumball. We also give an equivalent condition in terms of th…
Suppose that we have the unit Euclidean ball in $\R^n$ and construct new bodies using three operations - linear transformations, closure in the radial metric and multiplicative summation defined by $\|x\|_{K+_0L} = \sqrt{\|x\|_K\|x\|_L}.$ We prove that in dimension 3 this procedure gives all origin symmetric convex bodies, while this is no longer true in dimensions 4 and higher. We introduce the concept of embedding of a normed space in $L_0$ that naturally extends the corresponding properties of $L_p$-spaces with $p\ne0$, and show that the procedure described above gives exactly the unit ball…
Strictly convex space : Strong orthogonality and Conjugate diameters
本文研究了严格凸空间中的强正交性与共轭直径，证明了严格凸的等价条件，并给出了Hamel基成为强正交基、范数下界估计及共轭直径存在性的充要条件。
Generalized idempotents on the space of analytic functions with bounded derivatives
研究单位圆盘上有界导数解析函数空间上广义双循环幂等元的结构，通过定义相关映射并利用其性质进行分析。
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