On the Mazur--Ulam theorem in fuzzy n--normed strictly convex spaces

The Mazur-Ulam property for uniform algebras

We give a sufficient condition for a Banach space with which the homogeneous extension of a surjective isometry from the unit sphere of it onto another one is real-linear. The condition is satisfied by a uniform algebra and a certain extremely $C$-regular space of real-valued continuous functions.

The Mazur--Ulam property in $\ell_\infty$-sum and $c_0$-sum of strictly convex Banach spaces

In this paper we deal with those Banach spaces $Z$ which satisfy the Mazur--Ulam property, namely that every surjective isometry $Δ$ from the unit sphere of $Z$ to the unit sphere of any Banach space $Y$ admits an unique extension to a surjective real-linear isometry from $Z$ to $Y$. We prove that for every countable set $Γ$ with $\vert Γ\vert \geq 2$, the Banach space $\bigoplus_{γ\in Γ}^{c_0} X_γ$ satisfies the Mazur--Ulam property, whenever the Banach space $X_γ$ is strictly convex with dim$((X_γ)_{\mathbb{R}})\geq 2$ for every $γ$. Moreover we prove that the Banach space $C_0(K,X)$ satisfi…

Generalized-Lush Spaces and the Mazur-Ulam Property

We introduce a new class of Banach spaces, called generalized-lush spaces (GL-spaces for short), which contains almost-CL-spaces, separable lush spaces (specially, separable $C$-rich subspaces of $C(K)$), and even the two-dimensional space with hexagonal norm. We obtain that the space $C(K,E)$ of the vector-valued continuous functions is a GL-space whenever $E$ is, and show that the GL-space is stable under $c_0$-, $l_1$- and $l_\infty$-sums. As an application, we prove that the Mazur-Ulam property holds for a larger class of Banach spaces, called local-GL-spaces, including all lush spaces and…

Khintchine inequality on normed spaces and the application to Banach-Mazur distance

We establish variant Khintchine inequalities on normed spaces of Hanner type and cotype, in which the Rademacher distribution corresponding to classical Khintchine inequality is replaced by general symmetric distributions. The proof involves the $p$-barycenter and Birkhoff's ergodic theorem. More importantly, by employing these Khintchine inequalities, we get some lower bounds for Banach-Mazur distance between $l^p$-ball and a general centrally symmetric convex body.

Every commutative JB$^*$-triple satisfies the complex Mazur--Ulam property

We prove that every commutative JB$^*$-triple satisfies the complex Mazur--Ulam property. Thanks to the representation theory, we can identify commutative JB$^*$-triples as spaces of complex-valued continuous functions on a principal $\mathbb{T}$-bundle $L$ in the form $$C_0^\mathbb{T}(L):=\{a\in C_0(L):a(λt)=λa(t)\text{ for every } (λ,t)\in\mathbb{T}\times L\}.$$ We prove that every surjective isometry from the unit sphere of $C_0^\mathbb{T}(L)$ onto the unit sphere of any complex Banach space admits an extension to a surjective real linear isometry between the spaces.

A Note on The Mazur-Ulam Property of Almost-CL-spaces

We introduce the (T)-property, and prove that every Banach space with the (T)-property has the Mazur-Ulam property (briefly MUP). As its immediate applications, we obtain that almost-CL-spaces admitting a smooth point(specially, separable almost-CL-spaces) and a two-dimensional space whose unit sphere is a hexagon has the MUP. Furthermore, we discuss the stability of the spaces having the MUP by the $c_0$- and $\ell_1$-sums, and show that the space $C(K,X)$ of the vector-valued continuous functions has the the MUP, where $X$ is a separable almost-CL-space and $K$ is a compact metric space.

赋范空间的Banach-Mazur型

On the Banach-Mazur Type for Normed Spaces

为了测量定性性质，我们引入了一个概念的类型为任意赋范空间，措施的最坏可能增长的序列弱收敛到零的部分和。这些想法可以追溯到巴拿赫和马祖尔谁使用这种类型比较所谓的线性维数的经典巴拿赫空间。作为应用，我们比较了几个经典巴拿赫空间的线性维数并研究了它们的同构性.

非阿基米德赋范空间中的Mazur-Ulam定理

A Mazur--Ulam theorem in non-Archimedean normed spaces

经典的Mazur-Ulam定理指出，每个实赋范空间之间的满射等距是仿射的，这是不成立的非阿基米德赋范空间。本文在非阿基米德严格凸赋范空间中建立了Mazur-Ulam定理。

On the Mazur--Ulam property for the space of Hilbert-space-valued continuous functions

Let $K$ be a compact Hausdorff space and let $H$ be a real or complex Hilbert space with dim$(H_\mathbb{R})\geq 2$. We prove that the space $C(K,H)$ of all $H$-valued continuous functions on $K$, equipped with the supremum norm, satisfies the Mazur--Ulam property, that is, if $Y$ is any real Banach space, every surjective isometry $Δ$ from the unit sphere of $C(K,H)$ onto the unit sphere of $Y$ admits a unique extension to a surjective real linear isometry from $C(K,H)$ onto $Y$. Our strategy relies on the structure of $C(K)$-module of $C(K,H)$ and several results in JB$^*$-triple theory. For …

Every non-smooth $2$-dimensional Banach space has the Mazur-Ulam property

A Banach space $X$ has the $Mazur$-$Ulam$ $property$ if any isometry from the unit sphere of $X$ onto the unit sphere of any other Banach space $Y$ extends to a linear isometry of the Banach spaces $X,Y$. A Banach space $X$ is called $smooth$ if the unit ball has a unique supporting functional at each point of the unit sphere. We prove that each non-smooth 2-dimensional Banach space has the Mazur-Ulam property.

Corrigendum to "On the Mazur-Ulam theorem in non-Archimedean fuzzy anti-2-normed spaces"

In this note we correct a paper by D. Kang ("On the Mazur-Ulam theorem in non-Archimedean fuzzy anti-2-normed spaces", Filomat, 2017). The research in that paper applies to what the author calls strictly convex spaces. Nevertheless, we prove that this notion is void: there is no single space that satisfies the definition.

Every 2-dimensional Banach space has the Mazur-Ulam property

We prove that every isometry between the unit spheres of 2-dimensional Banach spaces extends to a linear isometry of the Banach spaces. This resolves the famous Tingley's problem in the class of 2-dimensional Banach spaces.

马祖尔-乌拉姆定理的概率泛化

A Probabilistic Generalization of the Mazur-Ulam Theorem

本文将马祖尔-乌拉姆定理泛化到概率空间，证明几乎处处保持距离的映射几乎处处与全局欧几里得等距映射一致。

The Mazur-Ulam property for the space of complex null sequences

Given an infinite set $Γ$, we prove that the space of complex null sequences $c_0(Γ)$ satisfies the Mazur-Ulam property, that is, for each Banach space $X$, every surjective isometry from the unit sphere of $c_0(Γ)$ onto the unit sphere of $X$ admits a (unique) extension to a surjective real linear isometry from $c_0(Γ)$ to $X$. We also prove that the same conclusion holds for the finite dimensional space $\ell_{\infty}^m$.

Mankiewicz's theorem and the Mazur--Ulam property for C*-algebras

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…

Approximation of a Cauchy-Jensen additive mapping in various normed spaces

In this paper, using the fixed point and direct methods, we prove the generalized Hyers-Ulam-Rassias stability of a Cauchy-Jensen additive functional equation in various normed spaces. The concept of Hyers-Ulam-Rassias stability originated from Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

The Mazur-Ulam property for commutative von Neumann algebras

Let $(Ω,μ)$ be a $σ$-finite measure space. Given a Banach space $X$, let the symbol $S(X)$ stand for the unit sphere of $X$. We prove that the space $L^{\infty} (Ω,μ)$ of all complex-valued measurable essentially bounded functions equipped with the essential supremum norm, satisfies the Mazur-Ulam property, that is, if $X$ is any complex Banach space, every surjective isometry $Δ: S(L^{\infty} (Ω,μ))\to S(X)$ admits an extension to a surjective real linear isometry $T: L^{\infty} (Ω,μ)\to X$. This conclusion is derived from a more general statement which assures that every surjective isometry …

Mazur-Ulam 定理与 Gromov-Hausdorff 距离

Mazur-Ulam Theorem With Gromov-Hausdorff Distance

本文在 Gromov-Hausdorff 距离框架下，研究了 Banach 空间线性等距的充要条件，并给出了有限维情况下的简化证明与等距判据。

Hilbert空间中闭线性关系的Hyers-Ulam稳定性

Hyers-Ulam stability of closed linear relations in Hilbert spaces

本文定义了线性关系的Hyers-Ulam稳定性，刻画了Hilbert空间中闭线性关系的稳定性，并给出了两个稳定线性关系的和与积保持稳定的充分条件。

A strengthened Kadison's transitivity theorem for unital JB$^*$-algebras with applications to the Mazur--Ulam property

The principal result in this note is a strengthened version of Kadison's transitivity theorem for unital JB$^*$-algebras, showing that for each minimal tripotent $e$ in the bidual, $\mathfrak{A}^{**}$, of a unital JB$^*$-algebra $\mathfrak{A}$, there exists a self-adjoint element $h$ in $\mathfrak{A}$ satisfying $e\leq \exp(ih)$, that is, $e$ is bounded by a unitary in the principal connected component of the unitary elements in $\mathfrak{A}$. This new result opens the way to attack new geometric results, for example, a Russo--Dye type theorem for maximal norm closed proper faces of the close…

广义陀螺向量空间再探

Generalized gyrovector spaces revisited

本文修正了广义陀螺向量空间中Mazur-Ulam定理的一个证明。

Voidness of strict convexity in non-Archimedean fuzzy normed spaces

In this short note, we show by elementary computations that the notion of non-Archimedean fuzzy normed (and 2-normed) spaces is void. Namely, there are no strictly convex spaces at all --not even the zero-dimensional linear space. Before this, we also study the case of strictly convex non-Archimedean normed spaces; in this setting we see that the only nonzero linear space (defined over an arbitrary non-Archimedean field) that satisfies this property is the one-dimensional linear space over $\mathbb{Z}/3\mathbb{Z}$. Consequently, the results that have been proven for this class of spaces, like …

Restricted Invertibility and the Banach-Mazur distance to the cube

We prove a normalized version of the restricted invertibility principle obtained by Spielman-Srivastava. Applying this result, we get a new proof of the proportional Dvoretzky-Rogers factorization theorem recovering the best current estimate. As a consequence, we also recover the best known estimate for the Banach-Mazur distance to the cube: the distance of every n-dimensional normed space from \ell_{\infty}^n is at most (2n)^(5/6). Finally, using tools from the work of Batson-Spielman-Srivastava, we give a new proof for a theorem of Kashin-Tzafriri on the norm of restricted matrices.

Phase-isometries on real normed spaces

We say that a mapping $f: X \rightarrow Y$ between two real normed spaces is a phase-isometry if it satisfies the functional equation \begin{eqnarray*} \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \quad (x,y\in X).\end{eqnarray*} A generalized Mazur-Ulam question is whether every surjective phase-isometry is a multiplication of a linear isometry and a map with range $\{-1, 1\}$. This assertion is also an extension of a fundamental statement in the mathematical description of quantum mechanics, Wigner's theorem to real normed spaces. In this paper, we show that for every space $Y$ the …

混合型加性-三次Jensen泛函方程在非阿基米德$(n, β)$赋范空间中的稳定性

Stability of Mixed Type Additive-Cubic Jensen Functional Equation in Non-Archimedean $(n, β)$ Normed Spaces

本文研究了混合型加性-三次Jensen泛函方程在非阿基米德$(n, β)$赋范空间中的Hyers-Ulam稳定性。

关于$\ell_\infty^n$的Banach--Mazur半径

On the Banach--Mazur radius of $\ell_\infty^n$

该研究证明了任意n维赋范空间与$\ell_\infty^n$的Banach--Mazur距离上界为$O(n^{2/3})$，推进了Banach--Mazur半径相关的几何泛函分析研究。

A Fixed Points Approach to stability of the Pexider Equation

Using the fixed point theorem we establish the Hyers-Ulam-Rassias stability of the generalized Pexider functional equation $$\frac{1}{\mid K\mid}\sum_{k\in K}f(x+k\cdot y)=g(x)+h(y),\;\;x,y\in E$$ from a normed space $E$ into a complete $β$-normed space $F$, where $K$ is a finite abelian subgroup of the automorphism group of the group $(E,+)$.

Perturbations of isometries between Banach spaces

We prove a very general theorem concerning the estimation of the expression $\|T(\frac{a+b}{2}) - \frac{Ta+Tb}{2}\|$ for different kinds of maps $T$ satisfying some general perurbated isometry condition. It can be seen as a quantitative generalization of the classical Mazur-Ulam theorem. The estimates improve the existing ones for bi-Lipschitz maps. As a consequence we also obtain a very simple proof of the result of Gevirtz which answers the Hyers-Ulam problem and we prove a non-linear generalization of the Banach-Stone theorem which improves the results of Jarosz and more recent results of D…

赋范空间的等边数的一个下界

A lower bound for the equilateral number of normed spaces

证明了如果n维赋范空间X与无限远之间的Banach-Mazur距离至多为3/2，则X中存在n +1个等距点.根据Alon和Milman的一个著名结果，这意味着任意n维赋范空间至少允许e ^{c sqrt（log n）}个等距点，其中c> 0是一个绝对常数。我们还证明了在足够接近n维ell p（1 <p <无穷大）的空间中存在n个等距点。

Polynomial estimates and radius of analyticity on real Banach spaces

A generalization of Problem 73 of Mazur and Orlicz in the Scottish Book was introduced from L. Harris. The exact value of the constant that appears there is known when complex normed linear spaces are considered. In this paper, we give estimates in the case of an arbitrary real normed linear space and a real $\ell_p$ space. Moreover, if $F(x)$ is a power series, $ρ$ its radius of uniform convergence and $ρ_{\substack{A}}$ its radius of analyticity, we prove that $ρ_{\substack{A}}\geqρ/\sqrt{2}$ and give some respective results for the $n$th Fréchet derivative of $F(x)$.

关于 Mazur 与 Sternbach 的一个问题

On a problem of Mazur and Sternbach

本文针对 Mazur–Sternbach 第155号问题，借助度量几何的新进展给出另一种证明路径，并在局部等距假设下得到肯定结论。

An Approximate Version of the Jordan von Neumann Theorem for Finite Dimensional Real Normed Spaces

It is known that any normed vector space which satisfies the parallelogram law is actually an inner product space. For finite dimensional normed vector spaces over R, we formulate an approximate version of this theorem: if a space approximately satisfies the parallelogram law, then it has a near isometry with Euclidean space. In other words, a small von Neumann Jordan constant E + 1 for X yields a small Banach-Mazur distance with R^n, d(X, R^n) < 1 + B_n E + O(E^2). Finally, we examine how this estimate worsens as the dimension, n, of X increases, with the conclusion that B_n grows quadratical…

向量格与赋范格上等距映射的刚性

Rigidity of isometries on vector and normed lattices

该研究探讨向量格与赋范格上等距映射的刚性，推广Baker定理，证明模等距、正范数等距的仿射性及相关性质，还证得非线性序-等距刚性定理，明确其不强制线性性。

赋范空间中近似凸集的大小

On the size of approximately convex sets in normed spaces

设X是赋范空间。X的子集A是近似凸的，如果对于所有的$a,b \in A$和$t \in [0,1]$都有$d(ta+(1-t)b,A) \le 1$，其中$d(x,A)$是$x$到$A$的距离。设$\Co(A)$为凸壳，$\diam(A)$为$A$的直径。本文证明了每个$n$维赋范空间都包含近似凸集$A$，其中$\mathcal{H}(A,\Co(A))\ge \log_2n-1$和$\diam(A) \le C\sqrt n(\ln n)^2$表示Hausdorff距离.这些估计是相当准确的。对于每个$D>0$，我们在$C[0,1]$中构造最坏可能的近似凸集，使得$\mathcal{H}(A,\Co(A))=\diam(A)=D$。还证明了与Hyers-Ulam稳定性定理有关的几个结果。

Banach-Mazur 距离和基常数的等距对称对数凹随机空间

Banach-Mazur distances and basis constants of isotropic log-concave random spaces

本文研究了由等距对称随机多面体生成的随机赋范空间间的Banach-Mazur距离，证明了在广泛参数范围内，该距离下界为cn/ln(1+m/n)，并扩展了Gluskin定理至一般等距对数凹测度，揭示了Banach-Mazur紧致集的极值几何现象。

非阿基米德空间中三次和四次函数方程的稳定性

Stability of cubic and quartic functional equations in non-Archimedean spaces

在非阿基米德赋范空间中，证明了三次函数方程$f(kx+y)+f(kx-y)=k[f(x+y)+f(x-y)]+2(k^3-k)f(x)$对所有$k\in \Bbb N$和四次函数方程$f(kx+y)+f(kx-y)=k^2[f(x+y)+f(x-y)]+2k^2(k^2-1)f(x)-2(k^2-1)f(y)$对所有$k\in \Bbb N$的广义Hyres-Ulam-Rassias稳定性.

The non-compact normed space of norms on a finite-dimensional Banach space

We discuss a new pseudometric on the space of all norms on a finite-dimensional vector space (or free module) $\mathbb{F}^k$, with $\mathbb{F}$ the real, complex, or quaternion numbers. This metric arises from the Lipschitz-equivalence of all norms on $\mathbb{F}^k$, and seems to be unexplored in the literature. We initiate the study of the associated quotient metric space, and show that it is complete, connected, and non-compact. In particular, the new topology is strictly coarser than that of the Banach-Mazur compactum. For example, for each $k \geqslant 2$ the metric subspace $\{ \| \cdot \…

关于F-空间中保持距离相等的映射

On maps which preserve equality of distance in F-spaces

为了推广Mazur-Ulam和Vogt的结果，我们证明了在没有满射条件的情况下，两个F-空间之间保持距离相等且T（0）=0的映射T是线性的.然后，作为特殊情况，线性等距通过其范围的简单性质来表征。

Stability of Euler-Lagrange type cubic functional equations in quasi-Banach spaces

In this paper, we study the generalized Hyers-Ulam stability of Euler-Lagrange type cubic functional equation of the form \begin{align*} 2mf(x + my) + 2f(mx - y) = (m^3 + m)[f(x+ y) + f(x - y)] + 2(m^4 - 1)f(y) \end{align*} for all $x,y \in X$, where $m$ is a fixed scalar such that $m \neq 0,1$, and $f$ is a map from a quasi-normed space $X$ to a quasi-Banach space $Y$ over the same field with $X$ by applying the alternative fixed point theorem.

某些ℓ_p求和与锥体的精确Banach-Mazur距离

Exact Banach-Mazur distances of certain $\ell_p$-sums and cones

本文研究了有限维实normed空间的ℓ_p直接和与相关凸体锥构造的Banach-Mazur距离，通过最近的最优Banach-Mazur位置特性，推导出从X_1⊕_p…⊕_pX_k到欧几里得空间的距离公式，并探讨了单锥与双锥的几何性质及距离关系。

Homogeneous Systems and Euclidean Topology

The Theorem on Invariance of Domain due to L.E.J. Brouwer states that one connected, compact (Hausdorff) m-dimensional manifold embedded into another actually realizes a homeomorphism. This fundamental result is relevant to Functional Analysis, as the classical Gelfand-Mazur Theorem, as well as the real form of the Fundamental Theorem of Algebra, can both be derived easily from it. Our main tool is the m-dimensional borsuk-Ulam Theorem: a certain real vector must be found, which is established by means of solving a real homogeneous system of equations as in the Theorem of Be'zout. We emphasize…

关于通过强三点齐次性刻画内积空间的一个注记

A remark on characterizing inner product spaces via strong three-point homogeneity

本文证明赋范线性空间等距同构于内积空间当且仅当它对任意（或每个）n≥3是强n点齐次度量空间，而n=2的情形对应Banach-Mazur问题。

Duality of gauges and symplectic forms in vector spaces

A gauge $γ$ in a vector space $X$ is a distance function given by the Minkowski functional associated to a convex body $K$ containing the origin in its interior. Thus, the outcoming concept of gauge spaces $(X, γ)$ extends that of finite dimensional real Banach spaces by simply neglecting the symmetry axiom (a viewpoint that Minkowski already had in mind). If the dimension of $X$ is even, then the fixation of a symplectic form yields an identification between $X$ and its dual space $X^*$ . The image of the polar body $K^{\circ}\subseteq X^*$ under this identification yields a (skew-)dual gauge…