Journal of Mathematical Analysis and Applications 235, 108᎐121Ž. 1999 Article ID jmaa.1999.6363, available online at http:rrwww.idealibrary.com on Properties of Isometric Mappings Themistocles M. Rassias Department of Mathematics, National Technical Uni¨ersity of Athens, Zografou Campus, 15780 Athens, Greece E-mail: [email protected] Submitted by William F. Ames Received February 9, 1999 DEDICATED TO THE MEMORY OF MOSHE FLATO IN ADMIRATION Some relations between isometry and linearity are examined. In particular, generalizations of the Mazur᎐Ulam theorem and open problems are discussed. ᮊ 1999 Academic Press Key Words: isometric mappings; Mazur᎐Ulam theorem; Banach spaces; strictly convex; topological vector spaces; approximate isometries; conservative distances. 1. INTRODUCTION The main theme of this article is the relation between isometry and linearity, that is, the Mazur᎐Ulam theorem and its generalizations. For a discussion of several other aspects of isometries on Banach spaces see the article by Fleming and Jamisonwx 19 . All vector spaces mentioned in this article are assumed to be real unless otherwise stated. Mazur and Ulam wx31 proved the following resultŽ see also Banachwx 5. THEOREM 1.1. Let U be an isometric transformation from a normed ¨ector space X onto a normed ¨ector space Y with UŽ.0 s 0. Then U is linear. Since continuity is implied by isometry it is only necessary to show that the isometric transformation U is additive, and additivity follows easily providing one can prove that U satisfies the functional equation q x12x 11 U s UxŽ.q Ux Ž.;1.1Ž. ž/2212 2 that is, U maps the midpoint of the line segment joining x12and x onto the midpoint of the line segment joining UxŽ.1212and Ux Ž.for all x and x 108 0022-247Xr99 $30.00 Copyright ᮊ 1999 by Academic Press All rights of reproduction in any form reserved. PROPERTIES OF ISOMETRIC MAPPINGS 109 in X. For the special case where both X and Y are strictly convex the proof ofŽ. 1.1 is immediate. A normed vector space E is called strictly con¨ex ŽClarksonwx 15. if, for each pair u, ¨ of nonzero elements in E such that 555555u q ¨ s u q ¨ , it follows that u s c¨ for some c ) 0. When X is strictly convex the only solution m of the pair of equations 5555y s y 555555y q y s y m x12x m and m x12x m x 21x Ž.1.2 s 11q s s is m 22x12x , with an analogous statement for y1UxŽ. 1and y 2 UxŽ.2 in Y. Thus, for strictly convex spaces the equationsŽ. 1.2 provide a metric characterization of the midpoint of the segment joining x12and x in X as well as that of the segment joining UxŽ.12and Ux Ž.in Y, so the linearity of U is demonstrated. A well-known example of a space which is not strictly convex is R2 with the norm 5Ž.a, b 5 s maxw<<<<a , b x. If we take s s x12Ž.0, 0 and x Ž.2, 0 in Ž. 1.2 , we find that every point on the line segment joining the pointsŽ.Ž. 1, y1 and 1, 1 is a solution of Ž. 1.2 . In order to prove that U is linear Mazur and Ulam needed to find a metric characterization of the midpoint of a pair of points that is valid for all normed spaces. They did this by constructing the following sequence of s sets Hm, m 1, 2, 3, . , where H1 is the set of solutions ofŽ. 1.2 ; thus s g 555555y s y s 1 y H11Ä4x X : x x x x22 x1x2 and s g 55y F 1 ␦ g HmmÄ4x H y1 : x z 2 Ž.Hmy1 for all z Hmy1 , m s 2,3,..., ␦ where Ž.Hmy1 designates the diameter of the set Hmy1. The intersection of these sets Hm is called the metric center of x12and x . Mazur and Ulam 11q proved that it consists of a single point 22x12x . In Section 2 we often deal with metric vector spaces. For these we use the following terminology. A real vector space with a metric dŽ.и, и satisfying duŽ.Ž.q ¨, w q ¨ s du, w for all u, ¨, w, and for which the operations of addition and scalar multiplication are jointly continuous will be called a metric ¨ector space. A complete metric vector space will be called an F-space, following BanachŽwx 5 , p. 35. It is often convenient to introduce the functional 55u s duŽ., 0 and call it the quasi-norm for the space. It clearly has the properties Ž.a 55u s 0 if and only if u s 0, Ž.b 555555u q ¨ F u q ¨ , Ž.1.3 Ž.c 5555y u s u , ␤ ª ␤ 55y ª 5␤ y ␤ 5ª Ž.d mmand u u 0 imply mumu 0 ¨ ␤ ␤ for u, , ummin the space and , real numbers. 110 THEMISTOCLES M. RASSIAS We note that the quasi-norm fails to be a norm since it is not required to be positive homogeneous. With the above definition of an F-space, local convexity is not required. We will use some standard definitions concern- ing topological vector spaces, including those of a semi-norm and a bounded set. A topological vector space is called locally bounded if every neighborhood of the origin contains a bounded open setŽ see Hyersw 22, 23x. 2. GENERALIZATIONS OF THE MAZUR᎐ULAM THEOREM The question of whether the Mazur᎐Ulam theorem holds for all metric vector spaces still seems to be open. The finite dimensional case was dealt with by Charzynskiwx 13 . He proved the theorem for a mapping U: E ª H with UŽ.0 s 0 which is an isometry between metric vector spaces E and H of the same finite dimension n. First he showed that it is sufficient to prove the theorem for the case H s E. Next, by a lengthy proof, he succeeded in defining a semi-norm sŽ.и on E which satisfies the condition sUwŽŽ.y Uz Ž..s sw Žy z .for all w, z in E. In the special case n s 1 the semi-norm is actually a norm, and the linearity of U follows from the Mazur᎐Ulam theorem. The rest of the proof is carried out by means of an induction on n. Rolewiczwx 44, 45 considered the class of metric vector spaces which are locally bounded and whose quasi-norms are conca¨e; that is, for each element u of the space, the function ␸: R ª R defined by ␸Ž.t s 55tu is concave for positive t. When both E and H are F-spaces subject to these two conditons, Rolewicz proved that each surjective isometry U: E ª H satisfying UŽ.0 s 0 is linear. Using both local boundedness and the concav- ity condition he first proved the following: LEMMA 2.1. Let E be a locally bounded F-space whose quasi-norm is ¨ s 555 5F ) conca e, and put mŽ. r supÄ u :2u r4. Then there exists r0 0 such - - - that mŽ. r r for 0 r r0. Rolewicz's proof now proceeds by a method analogous to that of Mazur᎐Ulam, but by first restricting points u and ¨ of E by the condition 55y ¨ - r u q ¨ ¨ u r0 2 in characterizing 22as the metric center of u and and later removing this restriction. A different kind of generalization of the Mazur᎐Ulam theorem was given by Daywx 16, pp. 110, 111 . He adapted their proof to the following situation. Let X and Y be locally convex topological vector spaces and let ª wx T: X Y carry a separating family ps of semi-norms to another such w X x X y s y family pssson Y by the rule pTxŽ.Ž.Ty px y , where T is surjec- tive and TŽ.0 s 0. Then T is linear. PROPERTIES OF ISOMETRIC MAPPINGS 111 In each of the theorems quoted above on the linearity of an isometry Ž.except Charzynski's , the hypothesis of surjectivity of the isometry was required. The necessity of this condition in general is shown by an example. Figielwx 18 cites the following. Let R have the usual absolute value as 2 5 5 s w<<<<x ª 2 norm and let R have the norm Ž.y12, y max y1, y 2, with T: R R given by TxŽ.s Žx, sin x .. Then T is an isometry of R into R2 with TŽ.0 s 0 which is not linear. Other examples are given by Bakerwx 4 , including that of a mapping F: R2 ª R3, where R2 has the Euclidean norm while the norm for R32is 5Ž.x, y, z 5 s maxÄ'x q y 2, <<z 4, such that F is a nonlinear isometry and is homogeneous of degree one. In the same paper Baker proves that if X and Y are normed vector spaces and Y is strictly convex, then every isometry T: X ª Y with TŽ.0 s 0 is linear. This provides another exception to the requirement of surjectivity in addition to Charzynski's theoremŽ see also Theorem 12, p. 145, of Aczel and Dhom- breswx 1. Wobstwx 50 generalized Charzynski's theorem as follows. THEOREM 2.2. Let E and F be metric ¨ector spaces with E finite dimen- sional. Let T: E ª F be a surjecti¨e isometry with TŽ.0 s 0. Then T is a linear mapping. He also proved that every isometry of a locally compact connected metric group with translation invariant metric into itself is surjective.