APPROXIMATE ISOMETRIES ON FINITE-DIMENSIONAL NORMED SPACES

APPROXIMATE ISOMETRIES ON FINITE-DIMENSIONAL NORMED SPACES S. J. DILWORTH Abstract. Every ε-isometry u between real normed spaces of the same finite dimension which maps the origin to the origin may by uniformly approximated to within 2ε by a linear isometry. Under a smoothness hypothesis, necessary and sufficient conditions are obtained for the same conclusion to hold for a given ε-isometry between infinite-dimensional Banach spaces. 1. Introduction The following notion of an approximate isometry between Banach spaces was introduced by Hyers and Ulam [5]. Definition 1. Let 0 <ε< and let E and F be real Banach spaces. A mapping f : E F is said to be an ε-isometry if f(x) f(y) x y ε (x, y E). Hyers and Ulam proved that every surjective ε-isometry on a Hilbert space may be uniformly approximated by an affine isometry. Recently [1] Bhatia and Semrl proved that every (not necessarily surjective) ε-isometry f from arealn-dimensional Euclidean space E n into itself, for which f(0) = 0, may be uniformly approximated to within 2ε by a linear isometry. The main purpose of this paper is to show that the same result holds for arbitrary finite-dimensional normed spaces, thus answering the question raised at the end of [1]. 2. Results Let E be a normed space. Recall that the norm of E is said to be Fréchet differentiable at y E if there exists a functional y E such that (1) y + h =1+y (h)+ h δ(h) (h E), where δ(h) 0as h 0. In fact y will be the unique norming functional for y. We shall write x y if y (x) = 0 (N.B. x y does not imply y x). Suppose that (x k ) k=1 is a bounded sequence in E (i.e. x k Cfor some fixed C) such that x k u for each k and that (λ k ) k=1 is a sequence of 1991 Mathematics Subject Classification. 46B04. 1

2 S. J. DILWORTH nonzero real numbers. The homogeneity of the norm function and (1) imply that λ k y + x k = λ k y +(1/λ k )x k = λ k (1 + ( x k / λ k )δ((1/ λ k )x k ) = λ k + x k δ((1/ λ k )x k ). In particular, if λ k as k, then (since x k C)wehave (2) λ k y + x k λ k 0ask. In the following proposition the hypothesis asserts that the set of points in E at which the norm is Fréchet differentiable should be dense. Among others, this hypothesis is satisfied by the Asplund spaces, which are by definition the spaces E for which every continuous convex function defined on an open subset U of E is Fréchet differentiable on a dense G δ subset of U (see e.g. [7, p.14]). It is known that a Banach space E is an Asplund space if and only if E has the Radon-Nikodým Property (equivalently, if and only if every separable subspace of E has a separable dual space) [7, p.34]. In particular, all reflexive Banach spaces are Asplund spaces. The hypothesis of the proposition is also satisfied by many spaces which are not Asplund spaces,e.g.byl. However, it is not satisfied by the important spaces l 1, L 1 [0, 1] and L [0, 1]. The existence of a dense set of points of Fréchet differentiability of the norm permits the argument of [1, p.503] to be generalized as follows. Proposition 1. Let E and F be normed spaces and suppose that the the set of points in E at which the norm is Fréchet differentiable is dense. Let f : E F be an ε-isometry, with f(0) = 0, for which there exist a constant K and a surjective linear isometry u : E F such that f(x) u(x) K for all x E. Then f(x) u(x) 2ε (x E). Proof. By replacing f by u 1 f, we may suppose without any loss of generality that F = E and that u = Id (the identity function), so that f(x) x Kfor all x E. Let x E and suppose that f(x) x = a 0. Fixα>0. By assumption there exists a unit vector y at which the norm is Fréchet differentiable such that f(x) x = ay + z, where z <α. Write x = x 0 + by (b R) withx 0 y. For every positive integer m we have f(x + my) =x+my + v m, where v m Kby hypothesis. Write v m = b m y + u m (b m R) withu m y. Let y be the unique norming functional at the point y. Then b m y = b m = y (b m y+u m ) = y (v m ) v m K, and so by the triangle inequality u m b m y + v m 2K. The fact that f is an ε-isometry with f(0) = 0 gives f(x + my) x+my ε,

APPROXIMATE ISOMETRIES ON FINITE-DIMENSIONAL NORMED SPACES 3 which can be rewritten as (m + b + b m )y +(x 0 +u m ) (m+b)y+x 0 ε. Since u m 2K, and since both x 0 y and u m y, the remarks before this lemma (in particular (2)) yield lim sup m+b+b m m+b ε. Thus lim sup b m ε. Using again the fact that f is an ε-isometry we have m ε f(x+my) f(x) m+ε, i.e. m ε (my + v m )+(x f(x)) m+ε, which can be rewritten as m ε (m a+b m )y+u m z m+ε. Since z <αthe triangle inequality gives m ε α< (m a+b m )y+u m <m+ε+α. Since u m 2Kand u m y, it follows from (2) that this norm can be made as close to m a + b m as we please by taking m sufficiently large. Since lim sup b m εit follows that a 2ε + α. Finally, since α can be chosen arbitrarily small, it follows that a 2ε. Remark 1. The surjectivity of u is essential. To see this, fix M > 0and ε>0 and consider f : R R 2 (with the Euclidean norm) given by { (x, 0) for x 0, f(x) = (x, M(1 e cx )) for x 0. Then f is an ε-isometry provided c = c(m,ε) is sufficiently small. However, sup{ f(x) (x, 0) : x R} = M, and so for large M the conclusion of the Proposition is not valid for the (non-surjective) isometry u(x) =(x, 0). Theorem 1. Let E and F be real normed spaces of the same finite dimension. Let f : E F be an ε-isometry from E into F with f(0) = 0. There exists a unique linear isometry u : E F such that u(x) f(x) 2ε (x E). Proof. The uniqueness of the linear isometry u is obvious and so it suffices to prove the existence of u. First, by a result attributed to P. L. Renz [2, Lemma 2.8], there exists a continuous 4ε-isometry g : E F such that g(0) = 0 and g(x) f(x) 2ε(x E). Secondly, by a result attributed to Dallas Webster [2, Prop. 4.1], whose proof contains an elegant application of the Borsuk-Ulam Theorem on antipodal points, every continuous approximate isometry between two normed spaces of the same finite dimension is surjective. Thus, g is a continuous surjective 4ε-isometry with g(0) = 0. By a result of Gruber [4, Theorem 3] there exists a surjective linear isometry u : E F such that u(x) g(x) 12ε. It follows from the triangle inequality that f(x) u(x) f(x) g(x) + g(x) u(x) 2ε+12ε=14ε.

4 S. J. DILWORTH As observed above, every finite-dimensional normed space satisfies the hypothesis of Proposition 1. In fact, it is a classical result that every continuous convex function on R n is differentiable almost everywhere with respect to n-dimensional Lebesgue measure (see e.g. [8, 7, Th. 25.5]). Finally, an application of Proposition 1 with K = 14ε yields the desired conclusion. Remark 2. It was shown in [1] that the upper bound of 2ε is best possible even in the case E = F = R. Here is a simple example; define f ε : R R thus: { x ε for x/ {0,ε}, f ε (x) = x for x {0,ε}. Clearly, f ε is a (surjective) ε-isometry, with f ε (0) = 0, and the identity function is the unique linear isometry which approximates f ε uniformly. But max{ f ε (x) x : x R} = f ε (ε) ε =2ε. The reader is referred to [6] for an example of a homeomorphism of R 2 for which the bound of 2ε is attained. An example is given in [5] (also [1]) which shows that the assumption that E and F should have the same dimension cannot be removed, even for Euclidean spaces, and that the result cannot be extended to infinite-dimensional spaces without some additional assumption. (See Theorem 2 below for a result of this kind.) Remark 3. For surjective ε-isometries Theorem 1 was proved by Gruber [4, Theorem 3] with the estimate 5ε. Hyers and Ulam [5] asked (implicitly) whether every surjective ε-isometry between normed spaces may be uniformly approximated by an isometry to within an error of Kε. This long-standing open question was finally answered positively by Gevirtz [3] with K = 5. Recently, Omladi c and Semrl [6] obtained the estimate K =2, which for linear approximation is optimal (see Remark 2 above). Corollary 1. Let E and F be real normed spaces of the same finite dimension and let f : E F be an ε-isometry from E into F. There exists an affine isometry v : E F such that v(x) f(x) 2ε (x E). Proof. Let g(x) = f(x) f(0) (x E). Then g is an ε-isometry with g(0) = 0. Theorem 1 applied to g gives a linear isometry u for which the affine isometry v(x) = u(x)+ f(0) is the required approximation to f. Some more terminology must now be introduced. A closed ball in a normed space with centre x and radius r will be denoted B(x, r). Recall that the Hausdorff distance between two subsets A and B of a metric space is defined by d H (A, B) =inf{r 0:A (B) r,b (A) r }, where (A) r = {x : d(x, A) <r}.letδ>0. A set S E will be said to be δ-dense in E if for every x E there exists s S such that x s δ;a mapping f : E F will be said to be δ-onto if f(e) isδ-dense in F.

APPROXIMATE ISOMETRIES ON FINITE-DIMENSIONAL NORMED SPACES 5 The next result is an infinite-dimensional version of Theorem 1. Under the hypothesis of Proposition 1 it provides necessary and sufficient conditions for the possibility of approximating an ε-isometry uniformly to within 2ε. Theorem 2. Let E and F be Banach spaces and suppose that the set of points in E at which the norm is Fréchet differentiable is dense. Let f : E F be an ε-isometry with f(0) = 0. Then the following are equivalent: (1) there exists a surjective linear isometry u : E F such that u(x) f(x) 2ε (x E); (2) f is δ-onto for some δ>0; (3) d H (f(e),f) <. Proof. Clearly, (2) and (3) are equivalent, and so it suffices to prove the equivalence of (1) and (2). Obviously, (1) (2) with δ =2ε. Suppose that (2) holds. Then the linear span of f(e) isdenseinf, which implies that the cardinality of E is not less than the cardinality of F. Let S = {x α : α A} Xbe a maximal set containing 0 satisfying the condition x α x β 2(α β). Then (2) and the fact that f is an ε-isometry readily imply that f(s) is(2+ε+δ)-dense in F. For each α Athere exists a surjection g α : B(x α, 1/2) B(f(x α ), 2+ε+δ) withg α (x α )=f(x α ). Define g : E F thus: { g α (x) for x B(x α,1/2), g(x) = f(x) otherwise. Then g is surjective and g(0) = 0. For x B(x α, 1/2), we have g(x) f(x) g α (x α ) f(x α ) + f(x α ) f(x) (2 + ε + δ)+(1/2+ε). Thus g(x) f(x) 5/2+2ε+δ (x E), and hence g is a (5 + 5ε +2δ)- isometry from E onto F. By a theorem of Gevirtz [3], there exists a surjective linear isometry u : E F with u(x) g(x) 5(5 + 5ε +2δ)(x E), which implies that u(x) f(x) K(x E) fork = 55/2 + 27ε + 11δ. Finally, (1) now follows from Proposition 1. The author does not know whether Theorem 2 holds without the hypothesis on E. For general Banach spaces, however, the following result affords necessary and sufficient conditions for the existence of a linear isometry which uniformly approximates a given ε-isometry. Proposition 2. Let f : E F be an ε-isometry between Banach spaces with f(0) = 0. The following are equivalent: (1) There exist K>0and a linear isometry u from E into F such that u(x) f(x) K (x E); (2) There exist M>0andaclosedsubspaceF 1 of F such that d H (f(e),f 1 )<M<.

6 S. J. DILWORTH Proof. If (1) holds then d H (u(e),f(e)) K, and so (2) holds with F 1 = u(e) andm=k. Suppose that (2) holds. Refining the proof of Theorem 2 (the details are omitted) one can construct a surjective mapping g : E F 1 satisfying g(0) = 0 and g(x) f(x) M+2ε (x E). Then g is a (2M +5ε)-isometry, and so by [6, Theorem 1] there exists a linear isometry from E onto F 1 satisfying u(x) g(x) 4M+10ε (x E). It follows that f satisfies (1) with K(M,ε) =5M+12ε. Remark 4. The example after Proposition 1 shows that (when F 1 F )the dependence of K(M,ε) onmcannot be eliminated and is at least linear in M. Acknowledgement. The author would like to thank Ralph Howard, Jim Roberts, Anton Schep and the members of the University of South Carolina Functional Analysis Seminar for comments which led to a simplification of the proof of Theorem 1. References [1] Rajendra Bhatia and Peter Semrl, Approximate Isometries on Euclidean Spaces, Amer. Math. Monthly, 104, 1997, 497 504. [2] R. D. BOURGIN, Approximate isometries on finite-dimensional Banach spaces, Trans. Amer. Math. Soc. 207 (1975) 309 328. [3] Julian Gevirtz, Stability of Isometries on Banach Spaces, Proc. Amer. Math. Soc., 89, 1983, 633 636. [4] P.M. Gruber, Stability of Isometries, Trans. Amer. Math. Soc., 245, 1978, 263 277. [5] D.H. Hyers and S.M. Ulam, On approximate isometries, Bull. Amer. Math. Soc., 51, 1945, 288 292. [6] M. Omladi c andp. Semrl On non linear perturbations of isometries, Math. Ann., 303, 1995, 617 628. [7] Robert R. Phelps, Convex functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, Vol. 1364, Springer-Verlag, Berlin Heidelberg, 1989. [8] R. Tyrell Rockafellar, Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970. Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A. E-mail address: dilworth@math.sc.edu