This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vectorvalued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysis.
The new methods developed here include a game approach to perturbational variational principles that is of independent interest. Detailed explanation of the underlying ideas and motivation behind the proofs of the new results on Fréchet differentiability of vectorvalued functions should make these arguments accessible to a wider audience. The most important special case of the differentiability results, that Lipschitz mappings from a Hilbert space into the plane have points of Fréchet differentiability, is given its own chapter with a proof that is independent of much of the work done to prove more general results. The book raises several open questions concerning its two main topics.

Front Matter Front Matter (pp. ivi) 
Table of Contents Table of Contents (pp. viix) 
Chapter One Introduction Chapter One Introduction (pp. 111)The notion of a derivative is one of the main tools used in analyzing various types of functions. For vectorvalued functions there are two main versions of derivatives: Gâteaux (or weak) derivatives and Fréchet (or strong) derivatives. For a function f from a Banach space X into a Banach space Y the Gâteaux derivative at a point
$x_0 \in X$ is by definition a bounded linear operator T : X → Y such that for every$u \in X$ ,$ \mathop {\lim }\limits_{t \to 0} \frac{{f(x_0 + tu)  f(x_0 )} } {t} = Tu. \caption{(1.1)} $ The operator T is called the Fréchet derivative of f at x_{0} if it is a Gâteaux derivative of f at x_{0} and...

Chapter Two Gâteaux differentiability of Lipschitz functions Chapter Two Gâteaux differentiability of Lipschitz functions (pp. 1222)We start by quickly recalling some basic notions and results that are well covered in [4]: the RadonNikodým property, main results on Gâteaux differentiability of Lipschitz functions, and related notions of null sets. We also discuss what is meant by validity of the mean value estimates, since this concept is deeply related to most of what is done in this book.
Lipschitz maps even from the real line into a Banach space need not have a single point of differentiability. A simple and wellknown example is f: [0, 1] → L_{1}([0, 1]) defined by
$ f(t) = 1_{[0,t]} . $ This map is even an...

Chapter Three Smoothness, convexity, porosity, and separable determination Chapter Three Smoothness, convexity, porosity, and separable determination (pp. 2345)In this chapter we prove some results that will be crucial in what follows; in particular, we show that spaces with separable dual admit a Fréchet smooth norm. For the first time, we meet the σporous sets and see their relevance for differentiability: the set of points of Fréchet nondifferentiability of continuous convex functions forms a σporous set. We also give some basic facts about σporous sets and show that they are contained in sets of Fréchet nondifferentiability of realvalued Lipschitz functions. These results, and their proofs, are important in order to understand some of the development that follows. So,...

Chapter Four εFréchet differentiability Chapter Four εFréchet differentiability (pp. 4671)In the context of all Fréchet differentiability results, or even of almost Fréchet differentiability ones, the results presented here are highly exceptional: they prove almost Fréchet differentiability in some situations when we know that the closed convex hull of all (even almost) Fréchet derivatives may be strictly smaller than the closed convex hull of the Gâteaux derivatives (see Chapter 14). Because of the possible future importance of this, so far only, foray into the otherwise impenetrable fortress of the problem of existence of derivatives in such situations, and because they have never appeared in book form before, we discuss the...

Chapter Five Γnull and Γ_{n}null sets Chapter Five Γnull and Γ_{n}null sets (pp. 7295)We define the notions of Γ and Γ_{n}null sets that will play a major role in our investigations of the interplay between differentiability, porosity, and smallness on curves or surfaces. Here we relate these notions to Gâteaux differentiability and investigate their basic properties. Somewhat unexpectedly, we discover an interesting relation between Γ and Γ_{n}null G_{δσ} sets, and this will turn out to be very useful in finding a new class of spaces for which the strong Fréchet differentiability result holds in Theorem 10.6.2.
In this chapter we introduce σideals of subsets of a Banach space X called Γnull sets or...

Chapter Six Fréchet differentiability except for Γnull sets Chapter Six Fréchet differentiability except for Γnull sets (pp. 96119)We give an account of the known genuinely infinite dimensional results proving Fréchet differentiability almost everywhere. This is where Γnull sets, porous sets, and special features of the geometry of the space enter the picture. Γnull sets provide the only notion of negligible sets with which a Fréchet differentiability result is known. Porous sets appear as sets at which Gâteaux derivatives can behave exceptionally badly (we call this behavior irregular), and they turn out to be the only obstacle to validity of a Fréchet differentiability result Γalmost everywhere. Finally, geometry of the space may (or may not) guarantee that porous...

Chapter Seven Variational principles Chapter Seven Variational principles (pp. 120132)Compared to direct recursive constructions such as those used in Chapter 15, the arguments of the following chapters will be significantly better organized and simplified by the use of variational principles (of Ekeland type). Our description of these principles as games and analysis following from it should also help to understand technical peculiarities of our arguments that stem from the careful order of the choice of parameters. In addition to the abstract variational principle, we also deduce some technical variants that will be used for special tasks later.
The goal of the variational principles that we consider in this short...

Chapter Eight Smoothness and asymptotic smoothness Chapter Eight Smoothness and asymptotic smoothness (pp. 133155)We introduce the smoothness notions that will be used to prove our main results: the modulus of smoothness of a function in the direction of a family of subspaces and the much simpler notion of upper Fréchet differentiability. This leads to the key notion of spaces admitting bump functions smooth in the direction of a family of subspaces with modulus controlled by ω(t). We show how this notion is related to asymptotic uniform smoothness, and that very smooth bumps, and very asymptotically uniformly smooth norms, exist in all asymptotically c_{0} spaces. This allows a new approach to results on Γalmost...

Chapter Nine Preliminaries to main results Chapter Nine Preliminaries to main results (pp. 156168)Most of this chapter revises some notions and results that will be used in subsequent chapters. In particular, we deepen the concept of regular differentiability and prove several inequalities controlling the increment of functions by the integral of their derivatives. The most important point of this chapter is the crucial lemma on deformation of ndimensional surfaces that will be basic in all results that we prove in the subsequent chapters. A number of results that should otherwise be here have already been used and therefore proved in the previous chapters, most notably the simple but important Corollary 4.2.9.
Much of...

Chapter Ten Porosity, Γ_{n} and Γnull sets Chapter Ten Porosity, Γ_{n} and Γnull sets (pp. 169201)In addition to the porosity notions that have been already defined, we introduce the notion of porosity “at infinity” (which we formally define as porosity with respect to a family of subspaces). Our main result shows that sets porous with respect to a family of subspaces are Γ_{n}null provided X admits a continuous bump function whose modulus of smoothness (in the direction of this family) is controlled by t^{n} log^{n1}(1/t). Corollaries include that in spaces with separable dual σporous sets are Γ_{1}null and, thanks to the logarithmic term, in Hilbert spaces they are Γ_{2}null. The first of these results is...

Chapter Eleven Porosity and εFréchet differentiability Chapter Eleven Porosity and εFréchet differentiability (pp. 202221)We show that every slice of the set of Gâteaux derivatives of a Lipschitz function f : X → Y , where dim Y = n, contains an εFréchet derivative provided certain porous sets associated with f are small in the sense that each of them can be covered by a union of Haar null sets and a Γ_{n}null G_{δ} set. By results of Chapter 10, this condition holds when X admits a bump function which is uniformly continuous on bounded sets and asymptotically smooth with modulus controlled by ω_{n}. In Chapter 13 we replace, under more restrictive assumptions, εFréchet...

Chapter Twelve Fréchet differentiability of realvalued functions Chapter Twelve Fréchet differentiability of realvalued functions (pp. 222261)We prove in this chapter that conemonotone functions on Asplund spaces have points of Fréchet differentiability, the appropriate version of the mean value estimates holds, and, moreover, the corresponding point of Fréchet differentiability may be found outside any given σporous set. This is a new result which considerably strengthens known Fréchet differentiability results for realvalued Lipschitz functions on such spaces. The avoidance of σporous sets is new even in the Lipschitz case. To explain the new ideas, in particular the use of the variational principle, and to introduce the reader to the proofs of more special but much harder differentiability...

Chapter Thirteen Fréchet differentiability of vectorvalued functions Chapter Thirteen Fréchet differentiability of vectorvalued functions (pp. 262318)We prove that if a space X admits a bump function which is upper Fréchet differentiable, Lipschitz on bounded sets, and asymptotically smooth with modulus controlled by t^{n} log^{n1}(1/t), then every Lipschitz map of X to a space of dimension not exceeding n has points of Fréchet differentiability. We also show that in this situation the multidimensional mean value estimate for Fréchet derivatives of locally Lipschitz maps of open subsets of X to spaces of dimension not exceeding n holds. In Chapter 14 we will see that this mean value statement is close to optimal. Particular situations in which our...

Chapter Fourteen Unavoidable porous sets and nondifferentiable maps Chapter Fourteen Unavoidable porous sets and nondifferentiable maps (pp. 319354)In Chapters 10 and 13 we have established conditions on a Banach space X under which porous sets in X are Γ_{n}null and/or the the multidimensional mean value estimates for Fréchet derivatives of Lipschitz maps into ndimensional spaces hold. Here we show in what sense these results are close to being optimal. Under conditions that can be argued to be close to complementary to those from the previous chapters, we find a σporous set whose complement is null on all ndimensional surfaces and the multidimensional mean value estimates fail even for εFréchet derivatives. Particular situations in which these negative results...

Chapter Fifteen Asymptotic Fréchet differentiability Chapter Fifteen Asymptotic Fréchet differentiability (pp. 355391)This chapter should be considered slightly experimental. We return to nonvariational arguments for proving differentiability, and try to construct the sequence converging to a point of differentiability by a less straightforward algorithm. The reason for this is the hope that a different algorithm can avoid the pitfalls indicated in Chapter 14 and prove, at least, that Lipschitz mappings of Hilbert spaces to finite dimensional spaces have points of Fréchet differentiability. From this point of view the results of this chapter are negative, although we provide a new proof of Corollary 13.1.2 on Fréchet differentiability of Lipschitz maps of Hilbert spaces...

Chapter Sixteen Differentiability of Lipschitz maps on Hilbert spaces Chapter Sixteen Differentiability of Lipschitz maps on Hilbert spaces (pp. 392414)For the benefit of those readers whose main interest is in Hilbert spaces, we give here a separate proof of existence of points of Fréchet differentiability of
$\mathbb{R}^2 $ valued Lipschitz maps on such spaces. Although the arguments are based on ideas from the previous chapters, only two technical lemmas whose proof may be easily read independently from the previous chapters are actually used. We also use this occasion to explain several ideas for treating the differentiability problem that may not have been apparent in the generality in which we have worked so far.We give here an essentially selfcontained proof...

Bibliography Bibliography (pp. 415418) 
Index Index (pp. 419422) 
Index of Notation Index of Notation (pp. 423425)