##
**Geometric nonlinear functional analysis. Volume 1.**
*(English)*
Zbl 0946.46002

Colloquium Publications. American Mathematical Society (AMS). 48. Providence, RI: American Mathematical Society (AMS). xi, 488 p. (2000).

The three classical tools of nonlinear analysis are topological methods, variational methods, and monotonicity methods. A detailed account of these methods may be found in E. Zeidler’s standard reference “Nonlinear functional analysis and its applications” [Vol. 1: Fixed point theorems (1993; Zbl 0794.47033); Vol. 2: Monotone operators (1990; Zbl 0684.47028/9); Vol. 3: Variational methods and optimization (1985; Zbl 0583.47051); Vol. 4: Applications to mathematical physics (1988; Zbl 0648.47036)]. Somewhat less is known about an equally important field, namely geometric aspects of nonlinear analysis, and no sufficiently detailed reference book has been available so far.

This deplorable gap is filled with the present book whose main theme is, according to the authors, the study of uniformly continuous and, in particular, Lipschitz functions between Banach spaces. The book consists of 17 chapters, several appendices on auxiliary topics, a bibliography containing about 600 items, and a very detailed and well-organized subject index. As it is common use now, each chapter closes with a section called “Notes and remarks”, where the authors also give several relevant bibliographical comments.

Since one can hardly describe the contents of this monograph better than the authors themselves, let us cite the corresponding part from the introduction. In Chapter 1 the authors consider uniformly continuous (and Lipschitz) retractions, extensions and selections. They prove that some important metric spaces are Lipschitz retracts of any metric space containing them. These include the space of continuous functions on compact metric spaces (with the usual sup norm) and the space of all closed, bounded and convex subsets of a given convex set in a Banach space (equipped with the Hausdorff metric). Then the authors study the question of extending uniformly continuous functions, defined on a subset of a Banach space \(E\) and taking values in a Banach space \(F,\) to the whole space \(E.\) This question is of special interest when \(E\) or \(F\) are Hilbert spaces. In the last section selections of set-valued maps are considered. The problems of retraction and of extension can both be formulated as selection problems, and the main interest here is in obtaining uniformly continuous selections whose moduli of continuity are as good as possible.

In Chapter 2 the authors consider, among others, three topics related to the material of the preceding chapter. The first is the question of approximating a general uniformly continuous function by functions with a prescribed modulus of continuity. In particular, they obtain the exact exponent \(\alpha\) so that every uniformly continuous function from the unit ball of \(L_p\) into \(L_q\) can be approximated arbitrarily well by functions of Hölder class \(\alpha.\) Another subject is the nearest point map onto convex sets in a uniformly convex and uniformly smooth Banach space. Here precise estimates on the modulus of continuity of this map are given. These estimates yield, in particular, that a space whose moduli of convexity and smoothness are of the best possible order must be isomorphic to a Hilbert space. The third topic is the Steiner point map, a classical method to select a distinguished point in a bounded convex subset in \(\mathbb R^n.\) The authors show, among other results, that its modulus of continuity on the set of closed bounded convex subsets in \(\mathbb R^n\) (equipped with the Euclidean norm) is the smallest possible among all such selection maps.

Chapter 3 is devoted to fixed point theory. Besides the basic standard results of Banach and Schauder, the authors also consider Lipschitz and nonexpansive maps of closed convex sets in Banach spaces. The result in this chapter which is most closely related to the rest of the book is that if \(C\) is a closed convex noncompact set in a Banach space, then there is a Lipschitz map \(f\) from \(C\) into itself with no approximate fixed point, i.e. \(\inf \{\|f (x) - x\|: x \in C\} > 0.\) In particular it follows that if \(E\) is an infinite-dimensional Banach space, then there is a Lipschitz retraction from the unit ball of \(E\) onto its unit sphere. The authors also discuss in some detail the rather delicate problem of the existence of fixed points and of the nonlinear ergodic theorem for nonexpansive maps.

One of the main tools of nonlinear functional analysis is the “linearization” of maps. A natural way to do so is to use derivatives. In the first section of Chapter 4 the authors present the two most natural concepts of differentiability, the Gâteaux and the Fréchet derivatives. They also present the basic calculus results that carry over with no difficulty from the finite-dimensional setting to infinite dimensions. The study of the existence of derivatives in infinite-dimensional spaces is much more involved than in finite-dimensional ones, and the main part of the chapter is devoted to the question of differentiation of convex real-valued functions. This subject is quite well understood today. There is even a complete characterization of sets in separable Banach spaces that can be sets of points of nondifferentiability (in the Gâteaux sense) of some convex continuous function.

Chapter 5 is concerned with Banach spaces which have the Radon-Nikodým property. This class of spaces appears naturally in a variety of contexts and has many equivalent characterizations involving, for example, vector-valued measures, vector-valued martingales, or the extremal structure of closed bounded convex sets. The characterization most relevant to the subject of this book is that spaces with Radon-Nikodým property are exactly those spaces \(E\) for which every Lipschitz function from \(\mathbb R\) to \(E\) is differentiable almost everywhere.

In Chapter 6 the authors turn to Gâteaux differentiability of Lipschitz functions. To study this topic they need, in addition to the concept of Radon-Nikodým spaces, also the notion of “almost everywhere” in infinite-dimensional Banach spaces. In spite of the fact that infinite-dimensional Banach spaces do not carry a natural measure, one can still define in a meaningful way the notion of negligible sets. Actually, this can be done in several nonequivalent ways. Some examples are given which illustrate these notions and also exhibit their delicate nature. With these tools (Radon-Nikodým property and negligible sets) at hand, the authors proceed to prove the existence almost everywhere of Gâteaux derivatives of Lipschitz functions between certain Banach spaces. This generalizes in a very useful way the classical theorem of Rademacher on the differentiability of Lipschitz functions between finite-dimensional spaces. In a separate section it is shown that the “almost everywhere” results do not hold for Fréchet differentiation even for real-valued convex and continuous functions on a separable Hilbert space. The chapter ends with a section which summarizes and gives an overview of the various classes of negligible sets which appear in this volume.

In Chapter 7 the Gâteaux differentiability results of Chapter 6 are applied in order to linearize Lipschitz retractions and bi-Lipschitz maps. This procedure reduces in many important cases the study of Lipschitz embeddings and retractions to the linear theory. It follows from the discussion in this chapter that some concrete separable spaces which are not linearly isomorphic to each other also fail to be Lipschitz equivalent. On the other hand, one may show that every separable Banach space is Lipschitz equivalent to a subset of \(c_0.\) This demonstrates the decisive role of the existence of derivatives in some of the results. The authors also present an example of two nonisomorphic spaces which are Lipschitz equivalent. (This example is nonseparable. No separable example of this type is known.) At the end of this chapter several examples of Lipschitz equivalences of \(\ell_2\) onto itself with some surprising properties are given.

Chapter 8 is devoted to uniform embeddings, i.e. uniformly continuous maps whose inverses are uniformly continuous on their domain of definition. In this chapter the authors study uniform embeddings into Hilbert space and give a complete linear characterization of the normed (even quasi-normed) spaces that embed uniformly into Hilbert space. They are exactly the spaces which are linearly isomorphic to a subspace of \(L_0 (\mu),\) the space of all \(\mu\)-measurable functions on some measure space with the topology of convergence in measure.

In Chapter 9 the authors characterize the Banach lattices whose unit spheres are uniformly homeomorphic to the unit sphere of Hilbert space. Examples (related to the Krivine-Maurey stability theory) show that for spaces which are not lattices the situation is more complicated. By using various methods (starting with the classical explicit map of Mazur, and proceeding via lattice factorization, complex interpolation and the Pelczyński decomposition method), the authors prove that the unit spheres of a very large class of spaces are uniformly homeomorphic to the unit sphere of a Hilbert space. They also show that the classification of unit balls of Banach spaces is essentially the same as the classification of their spheres.

Chapter 10 is concerned with uniform homeomorphisms between Banach spaces. It turns out that for several classes of Banach spaces, the structure of the space as a uniform space determines its linear structure. Moreover, for all Banach spaces the uniform structure already determines the finite-dimensional linear subspaces of the space (up to an isomorphism constant independent of the dimension of the subspace). Comparing these results with those of Chapter 9, one sees that there is a big difference between the uniform classification of spheres and that of spaces. The authors also give some examples that put the results of this chapter into perspective. For example, there are separable uniformly convex spaces which are uniformly homeomorphic but are not linearly isomorphic. There is even a separable uniformly convex space \(E\) with the following property: Any Banach space which is uniformly homeomorphic to \(E\) is either isomorphic to \(E\) or to \(E \oplus H\) (where \(H\) is an infinite-dimensional Hilbert space), and \(E\) and \(E \oplus H\) are uniformly homeomorphic but not linearly isomorphic. Another topic discussed in this chapter is the structure of discrete nets in Banach spaces. It is shown that knowing a discrete net in a Banach space up to Lipschitz equivalence gives much information on the space (and sometimes even determines it uniquely up to linear isomorphism). Infinite-dimensional Banach spaces determine their nets up to Lipschitz equivalence, but this is no longer true for finite-dimensional spaces (expect, of course, when the dimension is 1).

In Chapter 11 quotient maps are studied, a notion dual to that of embedding in the context of the linear theory. The direct analogue, namely, surjective Lipschitz or uniformly continuous maps, turns out to be insufficient from a general point of view. Indeed, for each pair of separable infinite-dimensional Banach spaces \(E\) and \(F\) there is a Lipschitz (and \(C^1\)) surjective map from \(E\) onto \(F.\) The authors introduce another notion of uniform (or Lipschitz) quotient map and study its properties. With this new notion they get, for example, that a uniform quotient space of a Hilbert space must be isomorphic to a Hilbert space. In finite dimensions the notion of Lipschitz quotient maps is related to the theory of quasi-regular maps.

In Chapter 12 the authors show that a real-valued uniformly continuous function on the unit sphere of a finite-dimensional Banach space is essentially constant on large sections of the sphere. More precisely, for every \(n,\) modulus of continuity \(\varphi,\) and \(\varepsilon > 0,\) there is a \(k = k (n, \varepsilon, \varphi)\) with \(k(n,\varepsilon,\varphi) \to \infty\) as \(n \to \infty\) (for fixed \(\varepsilon\) and \(\varphi\)), such that if \(f\) is a uniformly continuous real-valued function on the unit sphere of an \(n\)-dimensional Banach space \(E,\) and if the modulus of continuity of \(f\) is bounded by \(\varphi,\) then there is a \(k\)-dimensional subspace \(F\) of \(E\) such that the oscillation of \(f\) on the unit sphere of \(F\) is smaller than \(\varepsilon.\) The authors actually prove two results of this type. The first gives concrete estimates of \(k\) (which are sometimes asymptotically sharp), but does not show explicitly how \(F\) is located in \(E.\) The second gives much smaller \(k\) (and the authors do not even attempt to get any explicit estimate), but produces a subspace \(F\) which is nicely located in \(E\): If \(E\) has a basis, then \(F\) is spanned by a “block basis”. The heart of the first approach is the theorem of Dvoretzky on the existence of almost spherical sections in high-dimensional convex bodies. The second approach also reduces essentially to a result of a similar nature: For every \(k\) and \(\varepsilon\) there is an \(n\) such that every \(n\)-dimensional Banach space with a basis contains a block basis of length \(k\) which is \(\varepsilon\)-close to the unit vectors of \(\ell^k_p\) for some \(1\leq p\leq\infty.\)

In Chapter 13 the analogous question is discussed in an infinite-dimensional setup. The authors consider a uniformly continuous real-valued function \(f\) on the unit sphere of an infinite-dimensional Banach space \(E\) and ask whether there is an infinite-dimensional subspace \(F\) of \(E\) such that the restriction of \(f\) to the unit sphere of \(F\) has a small oscillation (prescribed in advance). The results of the previous chapter ensure the existence of finite-dimensional subspaces \(F\) of arbitrarily high dimension, but it turns out that in general there does not have to be an infinite-dimensional subspace \(F\) with the required property. The main example here is that of a distorted norm on a Hilbert space, i.e. an equivalent norm on a separable Hilbert space \(H\) (which one considers as a Lipschitz function on the unit sphere on \(H\)) for which there is a \(\delta > 0\) such that the oscillation of the new norm on the unit sphere of any infinite-dimensional subspace \(F\) of \(H\) is at least \(\delta.\) The construction and the proof are quite involved, and the norm is not constructed directly on a Hilbert space. The first step is the construction of a rather complicated Banach space. Results on the structure of this Banach space are then transferred to Hilbert space via an appropriate nonlinear map. The authors also show in this chapter that every uniformly continuous function from the unit sphere of \(c_0\) to \(\mathbb R\) has arbitrarily small oscillation when restricted to a suitable-dimensional subspace of \(c_0.\) It turns out that \(c_0\) is essentially the only Banach space with this property.

Chapter 14 considers maps which are close to isometries. After reviewing the main results concerning isometries, the authors present the theory developed by F. John in the framework of the theory of elasticity. Consider a map \(f\) defined on a bounded open set \(\Omega\) which is locally almost an isometry; then one asks whether \(f\) is injective on all of \(\Omega,\) or even close to an actual isometry there. Some of these questions can be answered for general Banach spaces, while others have a nice answer only in Hilbert space (actually only in finite-dimensional Euclidean spaces). In finite-dimensional spaces, the maps under consideration are differentiable almost everywhere, and it is natural to study their derivative. It is in the framework of this study that BMO functions were introduced for the first time.

Chapter 15 is concerned with the approximation of maps which are close to isometries by isometries, but an essential feature here is that the maps are globally defined and surjective. This issue is now well understood, and the authors present two quite precise results concerning it. chis study is closely related to that of functions which are almost linear in the sense that \(\|f(x+y)-f(x)-f(y)\|= o(\|x\|+\|y\|)\) as \(\|x\|, \|y\|\to \infty .\)

In Chapter 16 the authors study functions which are close to being linear in a weaker sense, i.e. \(\|f(x+y)-f(x)-f(y)\|= O(\|x\|+\|y\|).\) It turns out that these functions (the so-called quasi-linear functions) come up naturally in the study of “twisted sums” of Banach spaces. The natural setting for this study is not of Banach spaces, but that of quasi-Banach spaces. Quasi-linear functions are used in particular in order to produce nontrivial twisted sums of Hilbert space with itself. (The existence of such a twisted sum is far from obvious). The properties of twisted sums of Hilbert space with itself are studied in some detail. They give examples of Banach spaces which are interesting from many different points of view such as the study of inconditional structure or in the framework of complex Banach spaces, where they are used to produce two complex Banach spaces which are linearly isometric as real spaces but fail to be isomorphic as complex spaces.

Finally, in Chapter 17 the authors study questions related to Hilbert’s fifth problem (concerning Lie groups) in the setting of groups modeled on an infinite-dimensional Banach space. They first examine whether there exist nontrivial group structures on a Banach space. It is shown that under some metric and other natural conditions, the only way to put a commutative group structure on a Banach space is to transfer to it the addition operators from another Banach space via a uniform homeomorphism. The analogue of Hilbert’s problem concerning local groups is also considered. The authors study how far one can go in the construction of a Lie structure without changing the underlying model space. It is clear that without change of the model space one cannot solve the problem completely, and it seems to be unknown whether one can solve the infinite-dimensional analogue of Hilbert’s problem if a change in the model space is allowed. In all these considerations it is essential that one assumes that the group operation is uniformly continuous, and thus the subject is intimately connected to the uniform classification of Banach spaces.

This is the contents of the monograph as described by the authors in the introduction. Without any doubt, this is one of the great books on nonlinear analysis which will certainly become a standard reference. It is not only a must for every math library all over the world, but also for all researchers interested in functional analysis, operator theory, geometry of Banach spaces, and nonlinear problems. At the end of the introduction, the authors express their hope that this book will help in making the results of geometric nonlinear functional analysis better known to the general community of functional analysis, and that it will stimulate further research in the area. It definitely will.

This deplorable gap is filled with the present book whose main theme is, according to the authors, the study of uniformly continuous and, in particular, Lipschitz functions between Banach spaces. The book consists of 17 chapters, several appendices on auxiliary topics, a bibliography containing about 600 items, and a very detailed and well-organized subject index. As it is common use now, each chapter closes with a section called “Notes and remarks”, where the authors also give several relevant bibliographical comments.

Since one can hardly describe the contents of this monograph better than the authors themselves, let us cite the corresponding part from the introduction. In Chapter 1 the authors consider uniformly continuous (and Lipschitz) retractions, extensions and selections. They prove that some important metric spaces are Lipschitz retracts of any metric space containing them. These include the space of continuous functions on compact metric spaces (with the usual sup norm) and the space of all closed, bounded and convex subsets of a given convex set in a Banach space (equipped with the Hausdorff metric). Then the authors study the question of extending uniformly continuous functions, defined on a subset of a Banach space \(E\) and taking values in a Banach space \(F,\) to the whole space \(E.\) This question is of special interest when \(E\) or \(F\) are Hilbert spaces. In the last section selections of set-valued maps are considered. The problems of retraction and of extension can both be formulated as selection problems, and the main interest here is in obtaining uniformly continuous selections whose moduli of continuity are as good as possible.

In Chapter 2 the authors consider, among others, three topics related to the material of the preceding chapter. The first is the question of approximating a general uniformly continuous function by functions with a prescribed modulus of continuity. In particular, they obtain the exact exponent \(\alpha\) so that every uniformly continuous function from the unit ball of \(L_p\) into \(L_q\) can be approximated arbitrarily well by functions of Hölder class \(\alpha.\) Another subject is the nearest point map onto convex sets in a uniformly convex and uniformly smooth Banach space. Here precise estimates on the modulus of continuity of this map are given. These estimates yield, in particular, that a space whose moduli of convexity and smoothness are of the best possible order must be isomorphic to a Hilbert space. The third topic is the Steiner point map, a classical method to select a distinguished point in a bounded convex subset in \(\mathbb R^n.\) The authors show, among other results, that its modulus of continuity on the set of closed bounded convex subsets in \(\mathbb R^n\) (equipped with the Euclidean norm) is the smallest possible among all such selection maps.

Chapter 3 is devoted to fixed point theory. Besides the basic standard results of Banach and Schauder, the authors also consider Lipschitz and nonexpansive maps of closed convex sets in Banach spaces. The result in this chapter which is most closely related to the rest of the book is that if \(C\) is a closed convex noncompact set in a Banach space, then there is a Lipschitz map \(f\) from \(C\) into itself with no approximate fixed point, i.e. \(\inf \{\|f (x) - x\|: x \in C\} > 0.\) In particular it follows that if \(E\) is an infinite-dimensional Banach space, then there is a Lipschitz retraction from the unit ball of \(E\) onto its unit sphere. The authors also discuss in some detail the rather delicate problem of the existence of fixed points and of the nonlinear ergodic theorem for nonexpansive maps.

One of the main tools of nonlinear functional analysis is the “linearization” of maps. A natural way to do so is to use derivatives. In the first section of Chapter 4 the authors present the two most natural concepts of differentiability, the Gâteaux and the Fréchet derivatives. They also present the basic calculus results that carry over with no difficulty from the finite-dimensional setting to infinite dimensions. The study of the existence of derivatives in infinite-dimensional spaces is much more involved than in finite-dimensional ones, and the main part of the chapter is devoted to the question of differentiation of convex real-valued functions. This subject is quite well understood today. There is even a complete characterization of sets in separable Banach spaces that can be sets of points of nondifferentiability (in the Gâteaux sense) of some convex continuous function.

Chapter 5 is concerned with Banach spaces which have the Radon-Nikodým property. This class of spaces appears naturally in a variety of contexts and has many equivalent characterizations involving, for example, vector-valued measures, vector-valued martingales, or the extremal structure of closed bounded convex sets. The characterization most relevant to the subject of this book is that spaces with Radon-Nikodým property are exactly those spaces \(E\) for which every Lipschitz function from \(\mathbb R\) to \(E\) is differentiable almost everywhere.

In Chapter 6 the authors turn to Gâteaux differentiability of Lipschitz functions. To study this topic they need, in addition to the concept of Radon-Nikodým spaces, also the notion of “almost everywhere” in infinite-dimensional Banach spaces. In spite of the fact that infinite-dimensional Banach spaces do not carry a natural measure, one can still define in a meaningful way the notion of negligible sets. Actually, this can be done in several nonequivalent ways. Some examples are given which illustrate these notions and also exhibit their delicate nature. With these tools (Radon-Nikodým property and negligible sets) at hand, the authors proceed to prove the existence almost everywhere of Gâteaux derivatives of Lipschitz functions between certain Banach spaces. This generalizes in a very useful way the classical theorem of Rademacher on the differentiability of Lipschitz functions between finite-dimensional spaces. In a separate section it is shown that the “almost everywhere” results do not hold for Fréchet differentiation even for real-valued convex and continuous functions on a separable Hilbert space. The chapter ends with a section which summarizes and gives an overview of the various classes of negligible sets which appear in this volume.

In Chapter 7 the Gâteaux differentiability results of Chapter 6 are applied in order to linearize Lipschitz retractions and bi-Lipschitz maps. This procedure reduces in many important cases the study of Lipschitz embeddings and retractions to the linear theory. It follows from the discussion in this chapter that some concrete separable spaces which are not linearly isomorphic to each other also fail to be Lipschitz equivalent. On the other hand, one may show that every separable Banach space is Lipschitz equivalent to a subset of \(c_0.\) This demonstrates the decisive role of the existence of derivatives in some of the results. The authors also present an example of two nonisomorphic spaces which are Lipschitz equivalent. (This example is nonseparable. No separable example of this type is known.) At the end of this chapter several examples of Lipschitz equivalences of \(\ell_2\) onto itself with some surprising properties are given.

Chapter 8 is devoted to uniform embeddings, i.e. uniformly continuous maps whose inverses are uniformly continuous on their domain of definition. In this chapter the authors study uniform embeddings into Hilbert space and give a complete linear characterization of the normed (even quasi-normed) spaces that embed uniformly into Hilbert space. They are exactly the spaces which are linearly isomorphic to a subspace of \(L_0 (\mu),\) the space of all \(\mu\)-measurable functions on some measure space with the topology of convergence in measure.

In Chapter 9 the authors characterize the Banach lattices whose unit spheres are uniformly homeomorphic to the unit sphere of Hilbert space. Examples (related to the Krivine-Maurey stability theory) show that for spaces which are not lattices the situation is more complicated. By using various methods (starting with the classical explicit map of Mazur, and proceeding via lattice factorization, complex interpolation and the Pelczyński decomposition method), the authors prove that the unit spheres of a very large class of spaces are uniformly homeomorphic to the unit sphere of a Hilbert space. They also show that the classification of unit balls of Banach spaces is essentially the same as the classification of their spheres.

Chapter 10 is concerned with uniform homeomorphisms between Banach spaces. It turns out that for several classes of Banach spaces, the structure of the space as a uniform space determines its linear structure. Moreover, for all Banach spaces the uniform structure already determines the finite-dimensional linear subspaces of the space (up to an isomorphism constant independent of the dimension of the subspace). Comparing these results with those of Chapter 9, one sees that there is a big difference between the uniform classification of spheres and that of spaces. The authors also give some examples that put the results of this chapter into perspective. For example, there are separable uniformly convex spaces which are uniformly homeomorphic but are not linearly isomorphic. There is even a separable uniformly convex space \(E\) with the following property: Any Banach space which is uniformly homeomorphic to \(E\) is either isomorphic to \(E\) or to \(E \oplus H\) (where \(H\) is an infinite-dimensional Hilbert space), and \(E\) and \(E \oplus H\) are uniformly homeomorphic but not linearly isomorphic. Another topic discussed in this chapter is the structure of discrete nets in Banach spaces. It is shown that knowing a discrete net in a Banach space up to Lipschitz equivalence gives much information on the space (and sometimes even determines it uniquely up to linear isomorphism). Infinite-dimensional Banach spaces determine their nets up to Lipschitz equivalence, but this is no longer true for finite-dimensional spaces (expect, of course, when the dimension is 1).

In Chapter 11 quotient maps are studied, a notion dual to that of embedding in the context of the linear theory. The direct analogue, namely, surjective Lipschitz or uniformly continuous maps, turns out to be insufficient from a general point of view. Indeed, for each pair of separable infinite-dimensional Banach spaces \(E\) and \(F\) there is a Lipschitz (and \(C^1\)) surjective map from \(E\) onto \(F.\) The authors introduce another notion of uniform (or Lipschitz) quotient map and study its properties. With this new notion they get, for example, that a uniform quotient space of a Hilbert space must be isomorphic to a Hilbert space. In finite dimensions the notion of Lipschitz quotient maps is related to the theory of quasi-regular maps.

In Chapter 12 the authors show that a real-valued uniformly continuous function on the unit sphere of a finite-dimensional Banach space is essentially constant on large sections of the sphere. More precisely, for every \(n,\) modulus of continuity \(\varphi,\) and \(\varepsilon > 0,\) there is a \(k = k (n, \varepsilon, \varphi)\) with \(k(n,\varepsilon,\varphi) \to \infty\) as \(n \to \infty\) (for fixed \(\varepsilon\) and \(\varphi\)), such that if \(f\) is a uniformly continuous real-valued function on the unit sphere of an \(n\)-dimensional Banach space \(E,\) and if the modulus of continuity of \(f\) is bounded by \(\varphi,\) then there is a \(k\)-dimensional subspace \(F\) of \(E\) such that the oscillation of \(f\) on the unit sphere of \(F\) is smaller than \(\varepsilon.\) The authors actually prove two results of this type. The first gives concrete estimates of \(k\) (which are sometimes asymptotically sharp), but does not show explicitly how \(F\) is located in \(E.\) The second gives much smaller \(k\) (and the authors do not even attempt to get any explicit estimate), but produces a subspace \(F\) which is nicely located in \(E\): If \(E\) has a basis, then \(F\) is spanned by a “block basis”. The heart of the first approach is the theorem of Dvoretzky on the existence of almost spherical sections in high-dimensional convex bodies. The second approach also reduces essentially to a result of a similar nature: For every \(k\) and \(\varepsilon\) there is an \(n\) such that every \(n\)-dimensional Banach space with a basis contains a block basis of length \(k\) which is \(\varepsilon\)-close to the unit vectors of \(\ell^k_p\) for some \(1\leq p\leq\infty.\)

In Chapter 13 the analogous question is discussed in an infinite-dimensional setup. The authors consider a uniformly continuous real-valued function \(f\) on the unit sphere of an infinite-dimensional Banach space \(E\) and ask whether there is an infinite-dimensional subspace \(F\) of \(E\) such that the restriction of \(f\) to the unit sphere of \(F\) has a small oscillation (prescribed in advance). The results of the previous chapter ensure the existence of finite-dimensional subspaces \(F\) of arbitrarily high dimension, but it turns out that in general there does not have to be an infinite-dimensional subspace \(F\) with the required property. The main example here is that of a distorted norm on a Hilbert space, i.e. an equivalent norm on a separable Hilbert space \(H\) (which one considers as a Lipschitz function on the unit sphere on \(H\)) for which there is a \(\delta > 0\) such that the oscillation of the new norm on the unit sphere of any infinite-dimensional subspace \(F\) of \(H\) is at least \(\delta.\) The construction and the proof are quite involved, and the norm is not constructed directly on a Hilbert space. The first step is the construction of a rather complicated Banach space. Results on the structure of this Banach space are then transferred to Hilbert space via an appropriate nonlinear map. The authors also show in this chapter that every uniformly continuous function from the unit sphere of \(c_0\) to \(\mathbb R\) has arbitrarily small oscillation when restricted to a suitable-dimensional subspace of \(c_0.\) It turns out that \(c_0\) is essentially the only Banach space with this property.

Chapter 14 considers maps which are close to isometries. After reviewing the main results concerning isometries, the authors present the theory developed by F. John in the framework of the theory of elasticity. Consider a map \(f\) defined on a bounded open set \(\Omega\) which is locally almost an isometry; then one asks whether \(f\) is injective on all of \(\Omega,\) or even close to an actual isometry there. Some of these questions can be answered for general Banach spaces, while others have a nice answer only in Hilbert space (actually only in finite-dimensional Euclidean spaces). In finite-dimensional spaces, the maps under consideration are differentiable almost everywhere, and it is natural to study their derivative. It is in the framework of this study that BMO functions were introduced for the first time.

Chapter 15 is concerned with the approximation of maps which are close to isometries by isometries, but an essential feature here is that the maps are globally defined and surjective. This issue is now well understood, and the authors present two quite precise results concerning it. chis study is closely related to that of functions which are almost linear in the sense that \(\|f(x+y)-f(x)-f(y)\|= o(\|x\|+\|y\|)\) as \(\|x\|, \|y\|\to \infty .\)

In Chapter 16 the authors study functions which are close to being linear in a weaker sense, i.e. \(\|f(x+y)-f(x)-f(y)\|= O(\|x\|+\|y\|).\) It turns out that these functions (the so-called quasi-linear functions) come up naturally in the study of “twisted sums” of Banach spaces. The natural setting for this study is not of Banach spaces, but that of quasi-Banach spaces. Quasi-linear functions are used in particular in order to produce nontrivial twisted sums of Hilbert space with itself. (The existence of such a twisted sum is far from obvious). The properties of twisted sums of Hilbert space with itself are studied in some detail. They give examples of Banach spaces which are interesting from many different points of view such as the study of inconditional structure or in the framework of complex Banach spaces, where they are used to produce two complex Banach spaces which are linearly isometric as real spaces but fail to be isomorphic as complex spaces.

Finally, in Chapter 17 the authors study questions related to Hilbert’s fifth problem (concerning Lie groups) in the setting of groups modeled on an infinite-dimensional Banach space. They first examine whether there exist nontrivial group structures on a Banach space. It is shown that under some metric and other natural conditions, the only way to put a commutative group structure on a Banach space is to transfer to it the addition operators from another Banach space via a uniform homeomorphism. The analogue of Hilbert’s problem concerning local groups is also considered. The authors study how far one can go in the construction of a Lie structure without changing the underlying model space. It is clear that without change of the model space one cannot solve the problem completely, and it seems to be unknown whether one can solve the infinite-dimensional analogue of Hilbert’s problem if a change in the model space is allowed. In all these considerations it is essential that one assumes that the group operation is uniformly continuous, and thus the subject is intimately connected to the uniform classification of Banach spaces.

This is the contents of the monograph as described by the authors in the introduction. Without any doubt, this is one of the great books on nonlinear analysis which will certainly become a standard reference. It is not only a must for every math library all over the world, but also for all researchers interested in functional analysis, operator theory, geometry of Banach spaces, and nonlinear problems. At the end of the introduction, the authors express their hope that this book will help in making the results of geometric nonlinear functional analysis better known to the general community of functional analysis, and that it will stimulate further research in the area. It definitely will.

Reviewer: Jürgen Appell (Würzburg)

### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |