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# Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

Joram Lindenstrauss
David Preiss
Jaroslav Tišer
Pages: 424
Stable URL: http://www.jstor.org/stable/j.ctt7svpc

1. Front Matter (pp. i-vi)
3. Chapter One Introduction (pp. 1-11)

The notion of a derivative is one of the main tools used in analyzing various types of functions. For vector-valued functions there are two main versions of derivatives: Gâteaux (or weak) derivatives and Fréchet (or strong) derivatives. For a functionffrom a Banach spaceXinto a Banach spaceYthe Gâteaux derivative at a point$x_0 \in X$is by definition a bounded linear operatorT:XYsuch 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 operatorTis called the Fréchet derivative offatx0if it is a Gâteaux derivative offatx0and...

4. Chapter Two Gâteaux differentiability of Lipschitz functions (pp. 12-22)

We start by quickly recalling some basic notions and results that are well covered in [4]: the Radon-Nikodým 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 well-known example isf: [0, 1] →L1([0, 1]) defined by

$f(t) = 1_{[0,t]} .$

This map is even an...

5. Chapter Three Smoothness, convexity, porosity, and separable determination (pp. 23-45)

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 σ-porous sets and see their relevance for differentiability: the set of points of Fréchet nondifferentiability of continuous convex functions forms a σ-porous set. We also give some basic facts about σ-porous sets and show that they are contained in sets of Fréchet nondifferentiability of real-valued Lipschitz functions. These results, and their proofs, are important in order to understand some of the development that follows. So,...

6. Chapter Four ε-Fréchet differentiability (pp. 46-71)

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...

7. Chapter Five Γ-null and Γn-null sets (pp. 72-95)

We define the notions of Γ- and Γn-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 Γn-nullGδσsets, 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 σ-ideals of subsets of a Banach spaceXcalled Γ-null sets or...

8. Chapter Six Fréchet differentiability except for Γ-null sets (pp. 96-119)

We give an account of the known genuinely infinite dimensional results proving Fréchet differentiability almost everywhere. This is where Γ-null sets, porous sets, and special features of the geometry of the space enter the picture. Γ-null 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 Γ-almost everywhere. Finally, geometry of the space may (or may not) guarantee that porous...

9. Chapter Seven Variational principles (pp. 120-132)

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...

10. Chapter Eight Smoothness and asymptotic smoothness (pp. 133-155)

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 byω(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 asymptoticallyc0spaces. This allows a new approach to results on Γ-almost...

11. Chapter Nine Preliminaries to main results (pp. 156-168)

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 ofn-dimensional 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...

12. Chapter Ten Porosity, Γn- and Γ-null sets (pp. 169-201)

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 Γn-null providedXadmits a continuous bump function whose modulus of smoothness (in the direction of this family) is controlled bytnlogn-1(1/t). Corollaries include that in spaces with separable dual σ-porous sets are Γ1-null and, thanks to the logarithmic term, in Hilbert spaces they are Γ2-null. The first of these results is...

13. Chapter Eleven Porosity and ε-Fréchet differentiability (pp. 202-221)

We show that every slice of the set of Gâteaux derivatives of a Lipschitz functionf:XY, where dimY=n, contains anε-Fréchet derivative provided certain porous sets associated withfare small in the sense that each of them can be covered by a union of Haar null sets and a Γn-nullGδset. By results of Chapter 10, this condition holds whenXadmits a bump function which is uniformly continuous on bounded sets and asymptotically smooth with modulus controlled byωn. In Chapter 13 we replace, under more restrictive assumptions,ε-Fréchet...

14. Chapter Twelve Fréchet differentiability of real-valued functions (pp. 222-261)

We prove in this chapter that cone-monotone 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 σ-porous set. This is a new result which considerably strengthens known Fréchet differentiability results for real-valued Lipschitz functions on such spaces. The avoidance of σ-porous 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...

15. Chapter Thirteen Fréchet differentiability of vector-valued functions (pp. 262-318)

We prove that if a spaceXadmits a bump function which is upper Fréchet differentiable, Lipschitz on bounded sets, and asymptotically smooth with modulus controlled bytnlogn-1(1/t), then every Lipschitz map ofXto a space of dimension not exceedingnhas 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 ofXto spaces of dimension not exceedingnholds. In Chapter 14 we will see that this mean value statement is close to optimal. Particular situations in which our...

16. Chapter Fourteen Unavoidable porous sets and nondifferentiable maps (pp. 319-354)

In Chapters 10 and 13 we have established conditions on a Banach spaceXunder which porous sets inXare Γn-null and/or the the multidimensional mean value estimates for Fréchet derivatives of Lipschitz maps inton-dimensional 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 σ-porous set whose complement is null on alln-dimensional surfaces and the multidimensional mean value estimates fail even forε-Fréchet derivatives. Particular situations in which these negative results...

17. Chapter Fifteen Asymptotic Fréchet differentiability (pp. 355-391)

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...

18. Chapter Sixteen Differentiability of Lipschitz maps on Hilbert spaces (pp. 392-414)

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 self-contained proof...

19. Bibliography (pp. 415-418)
20. Index (pp. 419-422)
21. Index of Notation (pp. 423-425)