Mazur-Ulam Theorem Research Articles

Decomposition in direct sum of seminormed vector spaces and Mazur–Ulam theorem

Abstract It was proved by S. Mazur and S. Ulam in 1932 that every isometric surjection between normed real vector spaces is affine. We generalize the Mazur–Ulam theorem and find necessary and sufficient conditions under which distance-preserving mappings between seminormed real vector spaces are linear.

A set-valued extension of the Mazur–Ulam theorem

The Mazur–Ulam theorem states that every surjective isometry from a Banach space $X$ to a Banach space $Y$ is necessarily affine. Let $\mathfrak {K}(X)$ (resp. $\mathfrak {K}(Y)$) be the cone of all compact convex subsets of $X$ (resp. $Y$) endowed with t

A convex set-valued version of the Mazur-Ulam theorem on Asplund spaces

For a Banach space X, let C(X) be the cone consisting of all nonempty bounded closed convex subsets of X endowed with the Hausdorff metric. In this paper, we show that if one of the two Banach spaces X and Y is an Asplund space, then for every surjective isometry f:C(X)→C(Y), the restriction f|X is a surjective affine isometry from X to Y. If, in addition, one of X and Y is Fréchet smooth, or, locally uniformly convex, then f(C)=⋃{f(x):x∈C} for all C∈C(X).

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.

On the Mazur-Ulam theorem for Fréchet algebras

The Mazur-Ulam theorem for Fréchet algebras is proved.

Duality of gauges and symplectic forms in vector spaces

A gauge $$\gamma$$ 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 $$(X, \gamma )$$ 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 on X. In this paper, we study geometric properties of this so-called dual gauge, such as its behavior under isometries and its relation to orthogonality. A version of the Mazur–Ulam theorem for gauges is also proved. As an application of the theory, we show that closed characteristics of the boundary of a (smooth) convex body are optimal cases of a certain isoperimetric inequality.

On the Mazur–Ulam Theorem

We give an extension of the Mazur–Ulam theorem for isometries from real metrizable topological vector spaces into real normed spaces.

Mazur-Ulam theorem in probabilistic normed groups

In this paper, we give a probabilistic counterpart of Mazur-Ulam theorem in probabilistic normed groups. We show, under some conditions, that every surjective isometry between two probabilistic normed groups is a homomorphism.

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

We study the notion of a non-Archimedean fuzzy anti-2-normed space over a non-Archimedean field and prove that Mazur-Ulam theorem holds under some conditions in non-Archimedean fuzzy anti-2-normed spaces.

Examples and Applications of Generalized Gyrovector Spaces

In this paper we exhibit examples of generalized gyrovector spaces (GGVs) of the sets of positive invertible elements in unital C*-algebras. Then we employ a geometric inequality on the positive cones in these unital C*-algebras. By applying a Mazur–Ulam theorem for GGVs, we analyze Thompson-like isometries on GGVs of positive invertible elements.

Mazur-Ulam theorem for probabilistic 2-normed spaces

Mazur-Ulam theorem for probabilistic 2-normed spaces

On non- L 0 -linear perturbations of random isometries in random normed modules

The purpose of this paper is to study non- L 0 -linear perturbations of random isometries in random normed modules. Let (Ω,F,P) be a probability space, K the scalar field R of real numbers or C of complex numbers, L 0 (F,K) the equivalence classes of K-valued ℱ-measurable random variables on Ω, ( E 1 , ∥ ⋅ ∥ 1 ) and ( E 2 , ∥ ⋅ ∥ 2 ) random normed modules over K with base (Ω,F,P). In this paper, we first establish the Mazur-Ulam theorem in random normed modules. Making use of this theorem and the relations between random normed modules and classical normed spaces, we show that if f: E 1 → E 2 is a surjective random ε-isometry with f(0)=0 and has the local property, where ε∈ L 0 (F,R) and ε≥0, then there is a surjective L 0 -linear random isometry U: E 1 → E 2 such that ∥ f ( x ) − U ( x ) ∥ 2 ≤4ε, for all x∈ E 1 . We do not obtain a sharp estimate as the classical result, since random normed modules have a complicated stratification structure, which is the essential difference between random normed modules and classical normed spaces.MSC:46A22, 46A25, 46H25.

ON THE MAZUR-ULAM THEOREM IN FUZZY ANTI-2-NORMED SPACES

In this article, we study the notions of 2-isometries in fuzzy anti-2- normed spaces and prove a Mazur-Ulam type theorem in the 2-strictly convex fuzzy anti-2-normed spaces.

Isometry on linear n-normed spaces

This paper generalizes the Aleksandrov problem, the Mazur-Ulam theorem and Benz theorem on n-normed spaces. It proves that a one-distance preserving mapping is an n- isometry if and only if it has the zero-distance preserving property, and two kinds of n-isometries on n-normed spaces are equivalent.

On the Mazur-Ulam problem in fuzzy anti-normed spaces

The aim of this article is to proved a Mazur-Ulam type theorem in the strictly convex fuzzy anti-normed spaces. Keywords: Fuzzy anti-normed space, Mazur-Ulam theorem, strictly convex.

A Generalized Mazur-Ulam Theorem for Fuzzy Normed Spaces

We introducefuzzy norm-preservingmaps, which generalize the concept of fuzzy isometry. Based on the ideas from Vogt, 1973, and Väisälä, 2003, we provide the following generalized version of the Mazur-Ulam theorem in the fuzzy context: letX,Ybe fuzzy normed spaces and letf:X→Ybe a fuzzy norm-preserving surjection satisfyingf(0)=0. Thenfis additive.

On the Mazur-Ulam problem in non-Archimedean fuzzy 2-normed spaces

We study the notion of non-Archimedean fuzzy 2-normed space over a non-Archimedean field and prove that the Mazur-Ulam theorem holds under some conditions in the non-Archimedean fuzzy 2-normed space. MSC:46S10, 47S10, 26E30, 12J25.

On the Mazur-Ulam Theorem in Non-Archimedean Fuzzy -Normed Spaces

The motivation of this paper is to present a new notion of non-Archimedean fuzzy -normed space over a field with valuation. We obtain a Mazur-Ulam theorem for fuzzy -isometry mappings in the strictly convex non-Archimedean fuzzy -normed spaces. We also prove that the interior preserving mapping carries the barycenter of a triangle to the barycenter point of the corresponding triangle. And then, using this result, we get a Mazur-Ulam theorem for the interior preserving fuzzy -isometry mappings in non-Archimedean fuzzy -normed spaces over a linear ordered non-Archimedean field.

Mazur-Ulam theorem under weaker conditions in the framework of 2-fuzzy 2-normed linear spaces

The purpose of this paper is to prove that every 2-isometry without any other conditions from a fuzzy 2-normed linear space to another fuzzy 2-normed linear space is affine, and to give a new result of the Mazur-Ulam theorem for 2-isometry in the framework of 2-fuzzy 2-normed linear spaces. MSC: 03E72; 46B20; 51M25; 46B04; 46S40

A Mazur-Ulam problem in non-Archimedean n-normed spaces

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.MSC:46B20, 51M25, 46S10.