Boris Burshteyn, Alexander Volberg
Motivated by the questions in the theory of Fredholm stability in Banach space and Kato's strictly singular operators we answer several natural questions concerning ``orthogonality'' in normed spaces and the properties of metric projections. What the reader will see below might have benn known long ago, but we did not find it in the literature. Some open (for us) questions are formulated at the end of Sections 7 and 9.
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A complete characterization of Birkhoff-James orthogonality in infinite dimensional normed space
本文完整刻画了无限维实赋范线性空间上有界线性算子的Birkhoff-James正交性，并应用于有界线性泛函及算子光滑性的条件刻画。
On approximate Birkhoff-James orthogonality and normal cones in a normed space
本文从几何角度研究了赋范空间中近似 Birkhoff-James 正交性的两种概念，利用法锥对其进行刻画，并获得了二维光滑 Banach 空间中法锥的完整刻画及正交性集的唯一性定理。
Two different notions of approximate Birkhoff-James orthogonality in nor\-med linear spaces have been introduced by Dragomir and Chmie\-liń\-ski. In the present paper we consider a global and a local approximate symmetry of the Birkhoff-James orthogonality related to each of the two definitions. We prove that the considered orthogonality is approximately symmetric in the sense of Dragomir in all finite-dimensional Banach spaces. For the other case, we prove that for finite-dimensional polyhedral Banach spaces, the approximate symmetry of the orthogonality is equivalent to some newly introduced…
Let $X$ be a complex normed space. Based on the right norm derivative $ρ_{_{+}}$, we define a mapping $ρ_{_{\infty}}$ by \begin{equation*} ρ_{_{\infty}}(x,y) = \frac1π\int_0^{2π}e^{iθ}ρ_{_{+}}(x,e^{iθ}y)dθ\quad(x,y\in X). \end{equation*} The mapping $ρ_{_{\infty}}$ has a good response to some geometrical properties of $X$. For instance, we prove that $ρ_{_{\infty}}(x,y)=ρ_{_{\infty}}(y,x)$ for all $x, y \in X$ if and only if $X$ is an inner product space. In addition, we define a $ρ_{_{\infty}}$-orthogonality in $X$ and show that a linear mapping preserving $ρ_{_{\infty}}$-orthogonality has to…
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。
We introduce a notion of p-orthogonality in a general Banach space $1 \le p \le \infty$. We use this concept to characterize $\ell_p$-spaces among Banach spaces and also among complete order smooth p-normed spaces. We further introduce a notion of $p$-orthogonal decomposition in order smooth p-normed spaces. We prove that if the $\infty$-orthogonal decomposition holds in an order smooth $\infty$-normed space, then the 1-orthogonal decomposition holds in the dual space. We also give an example to show that the above said decomposition may not be unique.
On approximately orthogonality preserving and reversing operators
本文研究了近似正交保持与反转算子的性质，证明了某些条件下算子的正交保持性与单射性及ε-等距变换相关。
On directional preservation of orthogonality and its application to isometries
本文刻画线性算子沿指定方向对 Birkhoff-James 正交性的局部保持，并据此改进多面体赋范空间（含 $\ell_{\infty}^{n}$、$\ell_{1}^{n}$）上的等距识别定理。
Geometric properties of a novel type of orthogonality via norm derivatives
本文提出基于范数导数的线性组合的新型 $ρ_{α,β}$-正交性，研究其几何性质、光滑性判据及内积空间刻画，并证明保持该正交性的线性映射必为等距的标量倍数。
Orthogonality induced by norm derivatives : A new geometric constant and symmetry
本文引入新几何常数 $Γ(\mathbb{X})$ 研究 $ρ$-正交性与 Birkhoff-James 正交性的差异，探讨了空间几何性质与该常数的关系，刻画了 $ρ$-正交性的对称元素，并利用其对称性区分内积空间及描述特定 Banach 空间的对称元素。
Approximate Birkhoff-James orthogonality in the space of bounded linear operators
本文刻画了赋范空间上有界线性算子空间中两种近似Birkhoff-James正交性概念，并完整刻画了任意维数Hilbert空间上有界线性算子的近似Birkhoff-James正交性，推广了Chmieliński等人关于有限维Hilbert空间及紧算子的已有结果。
An application of the spherical completion to finite-dimensional normed spaces
本文利用球完备化建立有限维赋范空间的一般研究方法，借助正交性对非球完备域上的3维和4维赋范空间进行分类，并由此刻画严格epicompact集，解决一个公开问题。
Norming Approximate Orthogonality in Normed Linear Spaces
引入并研究赋范近似正交性，一种Birkhoff-James正交性的双参数推广，通过放松x的赋范条件和y的消失条件，建立了等价刻画，并统一了Chmieliński和Dragomir的近似正交概念。
The aim of this paper is to present some results concerning the $ρ_*$-orthogonality in real normed spaces and its preservation by linear operators. Among other things, we prove that if $T\,: X \longrightarrow Y$ is a nonzero linear $(I, ρ_*)$-orthogonality preserving mapping between real normed spaces, then $$\frac{1}{3}\|T\|\|x\|\leq\|Tx\|\leq 3[T]\|x\|, \qquad (x\in X)$$ where $[T]:=\inf\{\|Tx\|: \,x\in X, \|x\|=1\}$. We also show that the pair $(X,\perp_{ρ_*})$ is an orthogonality space in the sense of Rätz. Some characterizations of smooth spaces are given based on the $ρ_*$-orthogonality.
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。
We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approximate smoothness for some spaces. As an application, we consider in particular the Birkhoff-James orthogonality and its right-additivity under the assumption of approximate smoothness.
It is shown that Birkhoff-James orthogonality knows everything about the smooth norms in reflexive Banach spaces and can also compute the dimensions of the underlying normed spaces.
Associated to Birkhoff orthogonality, we study Birkhoff angles in a normed space and present some of their basic properties. We also discuss how to decide whether an angle is more acute or more obtuse than another. In addition, given two vectors $x$ and $y$ in a normed space, we study the formula for Birkhoff `cosine' of the angle from $x$ to $y$ from which we can, in principal, compute the angle. Some examples will be presented.
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。
Jordan Decomposition for WBV-functions in Ordered Normed Spaces
本文在有序赋范空间中引入基于正交性的强关系和基于有界线性泛函的弱关系，通过构造可扩张锥诱导弱关系，并据此证明了赋范空间中WBV函数的Jordan分解定理。
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.
We introduce the relation $ρ_λ$-orthogonality in the setting of normed spaces as an extension of some orthogonality relations based on norm derivatives, and present some of its essential properties. Among other things, we give a characterization of inner product spaces via the functional $ρ_λ$. Moreover, we consider a class of linear mappings preserving this new kind of orthogonality. In particular, we show that a linear mapping preserving $ρ_λ$-orthogonality has to be a similarity, that is, a scalar multiple of an isometry.
We study Birkhoff-James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and improve some of the recent results in this context. In particular, we obtain a characterization of Euclidean spaces and also prove that it is possible to retrieve the norm of a compact (bounded) linear operator (functional) in terms of its Birkhoff-James orthogonality set. We also present some best approximation type results in the space of bounded linear opera…
On the numerical radius parallelism and the numerical radius Birkhoff orthogonality
该研究将数值半径平行性与数值半径 Birkhoff 正交性推广至赋范空间算子，揭示其非传递、非加性等性质并给出刻画及二者联系。
Geometry of $\ell_p$-direct sums of normed linear spaces
研究可数多个赋范空间的 $\ell_p$-直和与 $c_0$-直和，通过对偶与支撑泛函刻画光滑性、近似光滑性及 Birkhoff-James 正交性。
Some inequalities and geometric constants in p-normed spaces
研究完全p-赋范空间中与等腰正交性相关的对称几何常数、Milman型模量及其不等式，并推广James常数与广义von Neumann-Jordan常数的关系。
We explore the relation between the orthogonality of bounded linear operators in the space of operators and that of elements in the ground space. To be precise, we study if $ T, A \in \mathbb{L}(\mathbb{X}, \mathbb{Y}) $ satisfy $ T \bot_B A,$ then whether there exists $ x \in \mathbb{X} $ such that $ Tx\bot_B Ax$ with $ \|x\| =1, \|Tx\| = \|T\|$, where $\mathbb{X}, \mathbb{Y} $ are normed linear spaces. In this context, we introduce the notion of Property $ P_n $ for a Banach space and illustrate its connection with orthogonality of a bounded linear operator between Banach spaces. We further …
Orthogonality and smoothness induced by the norm derivatives
本文研究赋范线性空间中由范数导数诱导的正交性与光滑性，利用对偶空间支撑泛函给出解析刻画，并建立局部光滑性与相应正交集的联系，通过紧算子空间等例子说明不同光滑性概念的异同。
Approximate Birkhoff-James orthogonality and smoothness in the space of bounded linear operators
本文研究赋范线性空间之间有界线性算子的近似Birkhoff-James正交性，并在紧算子空间为M-理想条件下刻画了算子的光滑性。
Birkhoff-James orthogonality and applications : A survey
本文综述 Birkhoff-James 正交性在 Banach 空间中的充要条件、几何应用及其与 Gateaux 导数和次微分的联系，并介绍相关距离公式与若干新结果。
On symmetricity of orthogonality with respect to numerical radius norm
该研究在数值半径型（半）范数下的向量值连续函数空间$C(K,X)$中，推广抽象数值域理论，刻画其左、右对称点，并得到紧算子空间$\n\nt\nh\ne \nK\n(X)$的对应结果。
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…
The concept of angle, angle functions, and the question how to measure angles present old and well-established mathematical topics referring to Euclidean space, and there exist also various extensions to non-Euclidean spaces of different types. In particular, it is very interesting to investigate or to combine (geometric) properties of possible concepts of angle functions and angle measures in finite-dimensional real Banach spaces (= Minkowski spaces). However, going into this direction one will observe that there is no monograph or survey reflecting the complete picture of the existing litera…
On Symmetry of Birkhoff-James Orthogonality of Linear Operators on Finite-dimensional Real Banach Spaces
本文刻画了有限维严格凸且光滑的实赋范线性空间上的左对称线性算子，证明了其当且仅当为零算子，并探讨了右对称线性算子的性质，证明了光滑线性算子不能是右对称的。
A complete characterization of smoothness in the space of bounded linear operators
本文利用 Birkhoff-James 正交性和半内积，通过范数化序列首次完整刻画了有界线性算子的光滑性，并证明在特定 Banach 空间条件下紧算子的 Gâteaux 可微性与 Fréchet 可微性等价。
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)。
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。
We prove that an asymmetric normed space is never a Baire space if the topology induced by the asymmetric norm is not equivalent to the topology of a norm. More precisely, we show that a biBanach asymmetric normed space is a Baire space if and only if it is isomorphic to its associated normed space.
In this paper, we study some topological characteristics of the n-normed spaces. We observe convergence sequences, closed sets, and bounded sets in the n-normed spaces using norms of quotient spaces that will be constructed. These norms will be a new viewpoint in observing the characteristics of the n-normed spaces. By using these norms, we also review the completeness of the n-normed spaces.
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.
In this paper, we introduce the cone normed spaces and cone bounded linear mappings. Among other things, we prove the Baire category theorem and the Banach--Steinhaus theorem in cone normed spaces.
We answer a question of Alex Koldobsky on isometric embeddings of finite dimensional normed spaces.
We generalize the notion of $M$-ideals in order smooth $\infty$-normed spaces to "smooth $p$-order ideals" in order smooth $p$-normed spaces. We show that if $V$ is an order smooth $p$-normed space and $W$ is a closed subspace of $V$, then $W$ is a smooth $p$-order ideal in $V$ if and only if $W^{\perp}$ is a smooth $p'$-order ideal in order smooth $p'$-normed space if and only if $W^{\perp\perp}$ is a smooth $p$-order ideal in order smooth $p$-normed space $V^{**}$. We prove that every $L$-summand in order smooth $1$-normed space is a smooth $1$-order ideal. We find a condition under which ev…
In this paper, we study main properties of cone normed spaces, and prove some theorems of weighted means in cone normed spaces.
Let X, Y be asymmetric normed spaces and Lc(X, Y) the convex cone of all linear continuous operators from X to Y. It is known that in general, Lc(X, Y) is not a vector space. The aim of this note is to prove, using the Baire category theorem, that if Lc(X, Y) is a vector space for some asymmetric normed space Y , then X is isomorphic to its associated normed space (the converse is true for every asymmetric normed space Y and is easy to establish). For this, we introduce an index of symmetry of the space X denoted c(X) $\in$ [0, 1] and we give the link between the index c(X) and the fact that L…
The idea of convex sets and various related results in 2-Probabilistic normed spaces were established in [HR]. In this paper, We obtain the concepts of convex series closedness, convex series compactness, boundedness and their interrelationships in Menger's 2-probabilistic normed space. Finally, the idea of $ \mathcal{D}- $ Boundedness in Menger's 2-probabilistic normed spaces and Menger's Generalized 2-Probabilistic Normed spaces are discussed.
In this paper, we establish the Mazur--Ulam theorem in the fuzzy strictly convex real normed spaces.
We consider linear narrow operators on lattice-normed spaces. We prove that, under mild assumptions, every finite rank linear operator is strictly narrow (before it was known that such operators are narrow). Then we show that every dominated, order continuous linear operator from a lattice-normed space over atomless vector lattice to an atomic lattice-normed space is order narrow.
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 …
In this paper, we study some features of n-normed spaces with respect to norms of its quotient spaces. We define continuous functions with respect to the norms of its quotient spaces and show that all types of continuity are equivalent. We also study contractive mappings on n- normed spaces using the same approach. In particular, we prove a fixed point theorem for contractive mappings on a closed and bounded set in an n-normed space.
We obtain an interesting inequalities for uniformly continuous functions in the normed spaces: $\|f(x)\|\leq a\|x\|+b$ for some $a,b> 0$.
The Erberlein-Smulian Theorem asserts that for complete normed spaces, that is Banach spaces, a subset is weak compact if and only if it is weak sequentially compact. In this paper it is shown that the completeness of the normed space is not necessary for the above mentioned result.
In this paper, we study a part of approximation theory that presents the conditions under which a closed set in a normed linear space is proximinal or Chebyshev.
In this paper, we generalize the Mazur--Ulam theorem in the fuzzy real n-normed strictly convex spaces.
In this paper we continue the study of compact-like operators in lattice normed spaces started recently by Aydin, Emelyanov, Erkurşun Özcand and Marabeh. We show among others, that every p-compact operator between lattice normed spaces is p-bounded. The paper contains answers of almost all questions asked by these authors.
In this article, we study the notions of $n$-isometries in non-Archimedean $n$-normed spaces over linear ordered non-Archimedean fields, and prove the Mazur-Ulam theorem in the spaces. Furthermore, we obtain some properties for $n$-isometries in non-Archimedean $n$-normed spaces.
In this paper we discuss some properties of resolvents of an accretive operator in linear 2-normed spaces, focusing on the concept of contraction mapping and the unique fixed point of contraction mappings in linear 2- normed spaces. Also, we establish the existence of solution of strong accretive operator equation in linear 2-normed spaces.
Inspired in the theorem of Krein-Milamn, we investigate the existence of extreme points in compact convex subsets of asymmetric normed spaces. We focus our attention in the finite dimensional case, giving a geometric description of all compact convex subsets of a finite dimensional asymmetric normed space.
We study metrically projective and metrically free matricially normed spaces. We describe these spaces in terms of a special space $\widehat M_n$, the space of $n\times n$ matrices, endowed with a special matrix-norm. We show that metrically free matricially normed spaces are matricial $\ell_1$-sums of some distinguished families of matricially normed spaces $\widehat M_n$, whereas metrically projective matricially normed spaces are complete direct summands of matricial $\ell_1$-sums of arbitrary families of the spaces $\widehat M_n$. At the end we specify the underlying normed space of $\wide…
The concept of Soft set theory was introduced by Molodtsov in the study [8]. Soft real numbers and properties were introduced inthe study [6] and soft normed space was defined in [11]. In this study, firstly we obtain a soft normed space by defining a soft norm on (Real numbers) which is called soft normed real space. By using this normed space we define the soft linear functional and investigate some of its properties. Secondly, we introduce soft dual space and soft dual operator and investigate their properties. Finally, we state and prove the theorem about representation of soft linear func…
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