Mazur-Ulam Theorem Research Articles

Mappings of Conservative Distances and the Mazur–Ulam Theorem

Let X and Y be two real Hilbert spaces with the dimension of X greater than 1. Several cases about the Aleksandrov–Rassias problem for T:X→Y preserving two or three distances are presented and geometric interpretations of these cases are also given.

Properties of Isometric Mappings

Some relations between isometry and linearity are examined. In particular, generalizations of the Mazur–Ulam theorem and open problems are discussed.

Approximate Isometries on Euclidean Spaces

This paper contains an exposition of two theorems on Banach spaces. Let X and Y be real Banach spaces and let f be a map from X to Y such that f(0)=0. The Mazur-Ulam Theorem says that if such a map is isometric (distance-preserving) and surjective, then it is linear. In general, it is necessary to assume that f is surjective. However, for a very large class of spaces this assumption is not necessary for the conclusion of the theorem. Let epsilon be a given positive number. An epsilon-isometry is a map from X to Y that preserves distances to within epsilon. It is known that if f is such a map between real Banach spaces, if f(0) = 0, and if f is surjective, then there exists a linear isometry g such that f and g are uniformly close. This was proved in 1945 by Hyers and Ulam for the special case of Hilbert spaces, and then extended to Banach spaces over the years by several authors. Here the assumption that f be surjective is necessary even when X and Y are Euclidean spaces.

Approximate Isometries on Euclidean Spaces

This paper contains an exposition of two theorems on Banach spaces. Let X and Y be real Banach spaces and let f be a map from X to Y such that f(0)=0. The Mazur-Ulam Theorem says that if such a map is isometric (distance-preserving) and surjective, then it is linear. In general, it is necessary to assume that f is surjective. However, for a very large class of spaces this assumption is not necessary for the conclusion of the theorem. Let epsilon be a given positive number. An epsilon-isometry is a map from X to Y that preserves distances to within epsilon. It is known that if f is such a map between real Banach spaces, if f(0) = 0, and if f is surjective, then there exists a linear isometry g such that f and g are uniformly close. This was proved in 1945 by Hyers and Ulam for the special case of Hilbert spaces, and then extended to Banach spaces over the years by several authors. Here the assumption that f be surjective is necessary even when X and Y are Euclidean spaces.

On the Mazur-Ulam Theorem and the Aleksandrov Problem for Unit Distance Preserving Mappings

Let $X$ and $Y$ be two real normed vector spaces. A mapping $f:X \to Y$ preserves unit distance in both directions iff for all $x,y \in X$ with $||x - y|| = 1$ it follows that $||f(x) - f(y)|| = 1$ and conversely. In this paper we shall study, instead of isometries, mappings satisfying the weaker assumption that they preserve unit distance in both directions. We shall prove that such mappings are not very far from being isometries. This problem was asked by A. D. Aleksandrov. The first classical result that characterizes isometries between normed real vector spaces goes back to S. Mazur and S. Ulam in 1932. We also obtain an extension of the Mazur-Ulam theorem.

On the Mazur-Ulam theorem and the Aleksandrov problem for unit distance preserving mappings

Let X X and Y Y be two real normed vector spaces. A mapping f : X → Y f:X \to Y preserves unit distance in both directions iff for all x , y ∈ X x,y \in X with | | x − y | | = 1 ||x - y|| = 1 it follows that | | f ( x ) − f ( y ) | | = 1 ||f(x) - f(y)|| = 1 and conversely. In this paper we shall study, instead of isometries, mappings satisfying the weaker assumption that they preserve unit distance in both directions. We shall prove that such mappings are not very far from being isometries. This problem was asked by A. D. Aleksandrov. The first classical result that characterizes isometries between normed real vector spaces goes back to S. Mazur and S. Ulam in 1932. We also obtain an extension of the Mazur-Ulam theorem.

Injectivity in Banach Spaces and the Mazur-Ulam Theorem on Isometries

Injectivity in Banach Spaces and the Mazur-Ulam Theorem on Isometries

Injectivity in Banach spaces and the Mazur-Ulam theorem on isometries

Injectivity in Banach spaces and the Mazur-Ulam theorem on isometries

A generalization of the Mazur-Ulam theorem

A generalization of the Mazur-Ulam theorem