ISOMETRIES AND APPROXIMATE ISOMETRIES

. Some properties of isometric mappings as well as approximate isometries are studied.

In general this is not easy. However, for the special case in which the space Y is strictly convex, the proof is simple (see Baker [1]). A (real) normed space is called strictly convex if, for each pair of its nonzero elements y, z such that y +z = y + z , it follows that y = cz for some real number c > 0. When Y is strictly convex, it is easy to show that the unique solution to the two equations is m = (y 1 /2) + (y 2 /2). Following Baker [1], we find that, from the second equation, since Y is strictly convex, m − y 1 = c(y 2 − m) with c > 0. From the first equation of (1.2), it follows that c = 1 and m = (y 1 /2) + (y 2 /2). Now, for any given pair x 1 , x 2 in X, let h = (x 1 + x 2 )/2 and put y 1 = U(x 1 ), y 2 = U(x 2 ). Clearly, we have Since U is an isometry, it follows that Hence, m = U (h) is the unique solution of (1.2), so that U is a solution of Jensen's equation (1.1). It follows that U(x)− U(0) is additive (cf. Aczél [9, page 43]).
In this strictly convex case, it was not necessary to assume that the mapping U was surjective. However, in general, the Mazur-Ulam Theorem 1.1 fails without this assumption. Among counterexamples in the literature which show this, we cite the following from Figiel [10]. Let Y denote the space of real number pairs y = (y 1 ,y 2 ) with the norm y = max[|y 1 |, |y 2 |], and consider the mapping T : R → Y given by T (s) = (s, sin(s)), s in R. Then, for s 1  so that T is an isometry but clearly is not additive. Of course, this space Y is far from being strictly convex. Indeed, if we take y 1 = (0, 0) and y 2 = (2, 0) in (1.2), we see that the solutions m to these equations form an infinite set, namely the line segment joining the points (1, −1) and (1,1). In their proof of Theorem 1.1, Mazur and Ulam found a way to produce a metric characterization of the midpoint of the segment joining two points y 1 and y 2 of an arbitrary normed vector space Y . They did this by constructing a sequence of sets H n in Y , n = 1, 2, 3,..., defined recursively by where δ(H n−1 ) denotes the diameter of the set H n−1 . The intersection of all the sets H n is called the metric center of y 1 and y 2 . They proved that it consists of a single point (y 1 /2) + (y 2 /2). Vogt [24] generalized Theorem 1.1 by considering mappings which preserve "equality of distance." These are mappings f : X → Y between normed vector spaces such that there exists a function p : [0, ∞) → [0, ∞) with f (x)−f (y) = p( x −y ). Such mappings were studied by Schoenberg [20] and by von Neumann and Schoenberg [25]. In Vogt's approach, a basic result was the following theorem concerning isometries in metric spaces. Proof. Let h : M → M be any surjective isometry. Since surjective isometries are injective, g −1 and h −1 exist and g, h, g −1 , h −1 are bijective isometries of M together with finite compositions of them. We define a sequence of isometries g n : M → M and a sequence of elements m n (n ∈ N) defined recursively by g n+1 = g n−1 g n (g n−1 ) −1 , m n+1 = g n−1 m n , n≥ 2.
Clearly, each g n is invertible on M. We now prove by induction that (1.9) By hypothesis, (1.9) is true for n = 1.
which demonstrates (1.9) for n = 2. Make the induction assumption that d(g n x, x) ≥ kd(m n ,x) when n ≥ 2. Then Proof of Theorem 1.1. Instead of using the original proof of Theorem 1.1 due to Mazur and Ulam, we shall rely on Vogt's method. This will "set the stage" for later developments when we deal with approximate isometries. Put (1.13) Put m = f (x) and let g : M → M be given by g(y) = 2f (x)− y. Then we see that (1.14) By Theorem 1.2 and properties (i)-(iv), we conclude that m is a fixed point of every surjective isometry h of M. In particular, we put x 0 = f −1 (2f (x)) and define h : (1.16) for all x ∈ X. Then f y : X → Y is a surjective isometry and f y (0) = 0, so that f y satisfies (vi). For any pair x, y in X, we have Thus, Comments: generalizations of the Mazur-Ulam theorem. As indicated above, Vogt used his method, that was used above to prove Theorem 1.1, together with other considerations in order to prove the following generalization. Theorem 1.3. Given real normed spaces X and Y , let f : X → Y with f (0) = 0 be a surjective mapping which preserves equality of distance. Then f is linear and f = βU, where β = 0 is a real number and U : X → Y is a surjective isometry, providing that the dimension of X is greater than one.
Vogt also gave an example to show that this theorem fails when X is one-dimensional.
We have seen that, in general, the isometry must be surjective in order to be linear. However, Figiel [10] proved the following result which amounts to another generalization of the Mazur-Ulam theorem. Theorem 1.4. Given real normed vector spaces X and Y , let ϕ : X → Y be an isometric embedding of X into Y . Then there exists a linear (not necessarily continuous) mapping F : Y → X such that F • ϕ is the identity on X and the restriction of F to the linear span of ϕ(X) is of norm one.
Various other generalizations or variations of the Mazur-Ulam theorem may be found in the literature. Charzyński [7] proved the theorem for metric vector spaces of the same finite dimension. Day [8, pages 110-111], demonstrated a version of the theorem involving semi-norms (instead of norms or metrics) in locally convex topological vector spaces. Rolewicz [19] proved the theorem for a class of metric vector spaces which are locally bounded and such that the function ϕ(t) = d(tx, 0) is concave in the real t for each fixed x in the space. An example of such a space is the set Wobst [26] generalized the results of Charzyński and of Rolewicz. To illustrate what can happen when an isometry is not surjective, we again turn to Baker [1] who gives two examples of isometries which are not affine. The first example is from Fischer and Muszély [11]: let Y be a real normed vector space which is not strictly convex. Thus, we may choose a, b in Y so that a and b are linearly independent and satisfy a = b = 1 and Using Baker's lemma, which states that a + b = a + b implies that sa + tb = s a +t b when s ≥ 0 and t ≥ 0, it is easy to see that f is an isometry. Also, f (0) = 0 and f is nonlinear.
The second example was constructed by Baker in answer to a problem of Chernoff (Advanced Problem 5688, Amer. Math. Monthly 76 (1969), 835). It shows that an isometry can be not only nonlinear but also homogeneous of degree one, and is defined as follows: let Y be the set R 3 but with the metric given by the norm (x,y,z) = (1.20) Clearly, g satisfies g(tx, ty) = tg(x, y) for real t, x, y, and it is easily seen that g is not linear. Also it can be shown that |g(x, 2 . Thus, f : R 2 → Y is an isometry which is nonlinear and homogeneous.

Approximate isometries: special cases.
By an ε-isometry of one metric space E into another E , we mean a mapping T : E → E which changes distances between elements of E by at most ε for some fixed ε > 0, that is, where d and d are the metrics for E and E , respectively. The stability question here is: given η > 0, does there exist ε > 0 and an isometry U : In the case, where E is a real Hilbert space and E = E , an answer was given in the affirmative by Hyers and Ulam [14], providing that T was surjective. Using the geometry of Hilbert space, they showed that, if T : (2.2a) They proved that the limit exists for all x in E and that U is an isometry. Finally, it was shown that, if the ε-isometry T is surjective, then U is also surjective and the inequality is satisfied for all x in E, with k = 10. This theorem was generalized by Bourgin [3], who obtained the results (2.2b) and (2.3) with k = 12 for an ε-isometry T of a Banach space E 1 onto a Banach space E 2 , where E 2 belongs to a class of uniformly convex spaces which includes the spaces L p (0, 1) for 1 < p < ∞.
Hyers and Ulam [15] studied ε-isometries for spaces of continuous functions. According to a famous theorem of Banach [2, page 170], two compact metric spaces S 1 , S 2 are homeomorphic if and only if the spaces C(S 1 ) and C(S 2 ) of real continuous functions defined on S 1 and S 2 are isometric. If h is a homeomorphism from S 1 onto S 2 , it is easily seen that C(S 1 ) and The hard part is to obtain the homeomorphism from the isometry.
Banach's theorem suggested a method of attacking the problem of stability for εisometries between spaces of continuous functions. Instead of using Banach's idea of peak functions, a different method of relating the ε-isometry to the correspondence between the points of S 1 and S 2 was used, by dealing with hyperplanes of functions having the same value at a given point. We shall outline the results without giving the details of the proofs. Lemma 2.1. Let S 1 and S 2 be compact metric spaces, and let T : Then, for each point p of S 1 , each b in R and each a ≥ 0, there exists a point q in S 2 and c in R such that This lemma was the first step in establishing a correspondence between points p of S 1 and q of S 2 . A corollary was proved to show that, when T is a surjective homeomorphism as well as an ε-isometry of C(S 1 ) onto C(S 2 ), then the point q of C(S 2 ) is uniquely determined by the point p of S 1 and is independent of the choice of the parameters b and a. Finally, the following stability result was proved.
Theorem 2.2. Let S 1 and S 2 be compact metric spaces. If T is a homeomorphic mapping of C(S 1 ) onto C(S 2 ) which is also an ε-isometry, then there exists an isometry Bourgin [4] obtained a significant generalization of Theorem 2.2 by adapting the basic idea of the proof outlined above to the more general case, with the following principal result. Here C(S j ) denotes the space of bounded continuous functions on S j , j = 1, 2.
Theorem 2.5. Let S 1 and S 2 be completely regular Hausdorff spaces and let T : Note. No continuity conditions were needed in Bourgin's theorem on the ε-isometry T , in contrast to the conditions of Theorem 2.2. His theorem also results in a generalization of the Banach-Stone theorem, again by replacing isometries by ε-isometries. More explicitly, he defined a µ-ideal as the subset of C(S 1 ) given by J(p, µ) = {ϕ ∈ C(S 1 ) : |ϕ(p)| < µ}, and similarly for C(S 2 ), and proved the following theorem.
Theorem 2.6. Let S 1 and S 2 be compact Hausdorff spaces. If H is a possibly multiplevalued mapping of C(S 1 ) onto C(S 2 ) and yields a correspondence of µ-ideals of either algebra to (µ + p)-ideals of the other, then S 1 and S 2 are homeomorphic.
3. Approximate isometries: the general case. The problem of stability of isometry for mappings between arbitrary real normed vector spaces remained open for many years after the work on special cases discussed in the previous section. In the present section, all normed spaces are assumed to be real.
The study of ε-isometries of Banach spaces was revived by Bourgin [6] and Bourgin [5], Omladič and Šemrl [18]. Gruber [13] demonstrated the stability of surjective isometries between all finite dimensional normed vector spaces. In addition, these authors obtained information concerning ε-isometries between arbitrary normed vector spaces which will be indispensible in the proof of the general case to be given below. In particular, the following lemma is a slight modification of Bourgin [6, Lemma 2.8].
Proof. Consider all subsets of X such that the distance between each pair of distinct elements is at least δ/2. By Zorn's lemma, there exists a maximal such collection which will be denoted by {x γ : γ ∈ Γ }. Then, for any x ∈ X, there is a γ ∈ Γ with x − x γ ≤ δ/2. Let B γ denote the ball {x ∈ X : x − x γ < δ}.
Bourgin [6] also introduced the following concept which also will be useful later.
Gruber [13] obtained an elegant and definitive result, as follows. The proof requires several lemmas subject to the hypotheses of Theorem 3.3.

Lemma 3.4. U is a surjective isometry.
Proof. From the hypotheses of the theorem, we have x / T x ≤ 1 + ε for large x , and uniformly as T (x) → ∞ uniformly. Hence, Y is the closure of the linear hull of U(X). Now we use Theorem 1.4 above to deduce the existence of a map J : Y → X, where J is linear and such that J • U = id X = identity on X, and J = 1. We must show that J is bijective. Assume that J is not injective. Then, because J is linear, there exists y = 0 in Y with J(y) = 0, so that J(βy) = 0 for all β ∈ R + . As was shown above,

T (x) − U(x) / T (x)
→ 0 uniformly as T (x) → ∞ uniformly. Therefore, we may choose for each β ∈ R + a point x β ∈ X such that βy − U(x β ) / βy and so also βy − U(x β ) /β → 0 as β → ∞. Recalling that J • U = id X and J = 1, we have This contradiction proves that J is injective. Since J • U = id X , J is also surjective and thus bijective.
Lemma 3.5. The isometry U : X → Y is given by the formula Proof. By the hypotheses of the theorem, given δ > 0, there exists M δ > 0 such that T (z)−U(z) ≤ δ z for all z ∈ X with z > M δ . For x = 0, equation (3.8) holds. For a given x = 0 in X, choose m so large that 2 m x ≥ M δ . Now U is linear, so, when n ≥ m, we have Given η > 0, choose δ < η/ x and we have By Lemmas 3.4, 3.5, and the fact that T (0) = 0, we find that: Now choose a mapping V −1 : X → X as follows. Put V −1 (0) = 0, and for x ∈ X with x = 0, let V −1 (x) be any point y ∈ X with V (y) = x. Then clearly V −1 has the following properties: For x ∈ X, by (3.11), (3.12), and (3.13), we have so that We now use Lemma 3.1 to approximate the ε-isometry V as follows. For each integer k > 0 let a continuous mapping V k : X → X be chosen so that (3.16) Lemma 3.6. Let y ∈ X and h in the dual space X * be chosen so that y = 1 and h = h(y) = 1. Then h(V (βy)) ≥ β − 4ε for each β > 0.
Proof. Denote the closed ball in X with center at zero and radius 1 by B. Let x ∈ X be chosen. Take y ∈ X with y = 1 and such that x −V (x) is an element of the halfray starting from the origin and containing y. Thus, x − V (x) = ωy, where ω ≥ 0. Next, choose u ∈ X such that y − u is a half-tangent to B at y, that is, and also that for suitable real numbers µ and θ, where θ ≥ 0. Thus, V (x) has the form V (x) = νy + θu (3.21) with ν = µ − ω ∈ R. We note that Denote by L the linear subspace generated by u and y. The line through y and y − u is a tac-line of B. Thus, the linear functional k : L → R defined by k(ξy +ηu) = ξ for ξ and η ∈ R, is of norm one. By the Hahn-Banach theorem, we can extend k to a linear functional h : X → R which also has norm one. Now h = h(y) = k(y) = y = 1, so by Lemma 3.6 we have h(V (βy)) ≥ β − 4ε with β > 0. Since V is an ε-isometry, we have Using (3.21) and noting that h(u) = k(u) = 0, we obtain the inequality On the other hand, by (3.20) for β > µ, we have By (3.19), given any η > 0, we may choose β = β(η) so large that βy − x ≤ β − µ + η.

Corollary 3.8. Under the hypotheses of Theorem 3.3, suppose that in addition T is continuous. Then the above inequality may be improved as U(x)− T (x) ≤ 3ε.
Proof. When T is continuous, we can eliminate the approximations V k and use V directly instead. Then the conclusion of Lemma 3.6 can be changed to h(V (βy)) ≥ β − 2ε, and the corollary follows.
It remained for Gevirtz [12] to at last prove the following result which establishes the stability of isometries between arbitrary Banach spaces. Theorem 3.9. Given real Banach spaces X and Y , let f : X → Y be a surjective εisometry. Then there exists a surjective isometry U : The main part of the proof is to demonstrate the following result. Theorem 3.10 (Gevirtz [12]). If f : X → Y is a surjective ε-isometry, then, for any pair x 0 ,x 1 ∈ X, we have Proof. The idea of the proof was to "epsilonize" the method of Vogt which was used above in demonstrating Theorems 1.1 and 1.2. From Bourgin's δ-onto idea, Gevirtz developed the following definition. for all y ∈ Y is called a δ-inverse of f and f is called δ-onto if it has a δ-inverse. Also, the term (δ, ε)-isometry is an abbreviation for "δ-onto ε-isometry." We will need the following lemmas.
(3.44) Now use the induction hypothesis with y replaced by G n−1 (y) to find that (3.45) so the induction is complete for (3.43).
(3.47) Now, by (3.42), a n − a n−1 ≤ 2 d + 4 n ε , (3.48) and we have a 2 − a 1 ≤ 2 2−n 2d + 2 4 n ε + 4 n−1 ε for n ≥ 2. (3.49) Changing n to n + 2, now for n ≥ 0, we find that By (3.34) and (3.40), a 1 = q and a 2 = g 0 (q) = f (2p − F(q)). Hence, We consider two cases according to whether or not d > 18ε. 18ε). Let the real number t be defined by d2 −t = 18ε2 t , that is, t = log 4 (d/18ε) > 0. Let n be the greatest integer in t. Then 2 −n < 2 · 2 −t so by (3.52) (3.53) by the definition of t. Note also from this definition that Since f is an ε-isometry and d > 18ε, we have ). This case covers the situation, where y 0 = y 1 which was excluded earlier. Here so that Thus, x j − p ≤ 19ε and y j − f (p) ≤ 20ε, j = 0, 1. Since (3.61) Therefore, in either Case 1 or Case 2, we have demonstrated the inequality Proof of Theorem 3.9. We may assume, without loss of generality, that f (0) = 0. Put x 0 = 2x, x 1 = 0 in (3.26) to get We prove by induction that By replacing x with 2 m x in (3.65) and dividing the result by 2 m , we get which shows that (2 −n f (2 n x)) is a Cauchy sequence. Since Y is a Banach space, there exists U : X → Y such that Moreover, if n → ∞ in (3.65), we obtain It follows immediately that U(x) − f (x) / x → 0 uniformly as x → 0 uniformly, and we know that f (0) = U(0) = 0. In order to apply Theorem 3.3, it remains only to show that U is an isometry. Since f is an ε-isometry on X, we have and in the limit as n → ∞ we get U(x 0 ) − U(x 1 ) = x 0 − x 1 for every pair of points x 0 ,x 1 ∈ X. Now all the requirements of the Gruber Theorem 3.3 above are satisfied and we conclude that U is surjective and that U(x)− f (x) ≤ 5ε for all x ∈ X.
Comments. Gruber [13] raised the question whether his bounds 3ε for T (x) − U(x) when the ε-isometry T is continuous and 5ε in general could be improved. He showed that if, in Theorem 3.3 above, the space X is Euclidean, then the bounds 3ε and 5ε may be replaced by 3ε/ √ 2 and 5ε/ √ 2, respectively. Recently, Omladič and Šemrl [18] have shown that indeed these bounds can be reduced. Their main result is the following theorem. Theorem 3.15. Let X and Y be real Banach spaces. Given ε > 0, let f : X → Y be a surjective ε-isometry with f (0) = 0. Then there exists a surjective isometry U : (3.70) These authors gave examples to show that 2ε is the best possible result. Their simplest example is this. Define f : R → R by It is easy to see that f is a 1-isometry, U(t) = t for this function, and also Thus, the inequality is sharp. They also constructed an example of a homeomorphic mapping f : R 2 → R 2 which is a 1-isometry and such that 2 cannot be replaced by a smaller constant in (3.70). Thus, the sharpness persists even for homeomorphisms.
Skof [21] has given a different approach to the stability problem for isometric mappings between real normed spaces X and E, in which neither completeness of the spaces nor surjectivity of the mappings is required. Also, the methods used are elementary.
Skof's theorem. Let T : X → E with T (0) = 0 satisfy the conditions: (1) T is a δisometry for some δ > 0, and (2) for each x ∈ X, both T (−x) and T (2x) are scalar multiples of T (x). Let E 0 denote the linear hull of T (X) ∈ E. Then there exists an isometry U : X → E 0 which satisfies the inequality T (x)− U(x) < 36δ for all x ∈ X and is given by the formula U(x) = lim n→∞ 2 −n T (2 n x).
The proof depends upon another theorem by the same author which establishes the stability of the alternative Cauchy equation: ϕ(x + y) = ϕ(x) + ϕ(y) . In general, neither T nor U need to be linear. For instance, recall the example given by Baker [1] which was mentioned after Theorem 3.15. Let X = R 2 with the usual norm, let E be R 3 but with the norm (x,y,z) = max x 2 + y 2 , |z| and f : X → E be the mapping given by Baker in the second example cited above. Then f is an isometry which is homogeneous and nonlinear. Clearly, this mapping f satisfies the conditions of Skof's theorem. Here T = U = f . ε-isometries on bounded sets. Swain [23] considered ε-isometries on bounded metric spaces and proved the following result. The same author also gave an example to show that even for ε-isometries T on convex subsets M of R 2 into R 2 there is no universal constant k such that d(T (x), U(x)) ≤ kε for x ∈ M and for all such subsets M. The difficulty arises here for long thin sets M.
The problem of ε-isometries on bounded subsets of R n was studied by Fickett [9]. He proved the following two stability theorems, the first depending on the size of the subset in question and the second on the shape of the subset.
The next theorem of Fickett illustrates the "difficulty" referred to above for thin sets. He defines the minimum thickness of a bounded set S to be the infimum of those d > 0 such that S will fit between two parallel hyper-planes a distance d apart. Fickett used the first of these results to prove a theorem concerning a conjecture of Ulam on the invariance of measure in the Hilbert cube.
Jung [16] modified the method (3.8) slightly to prove the stability of isometries on restricted but unbounded domains, while Fickett investigated the stability problems of isometries on bounded sets.