<br><h3> Chapter One </h3> <b>Introduction</b> <p> <p> 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 function <i>f</i> from a Banach space <i>X</i> into a Banach space <i>Y</i> the Gâteaux derivative at a point <i>x</i><sub>0</sub> [member of] <i>X</i> is by definition a bounded linear operator <i>T : X -> Y</i> such that for every <i>u [member of] X</i>, <p> [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1) <p> The operator <i>T</i> is called the Fréchet derivative of <i>f</i> at <i>x</i><sub>0</sub> if it is a Gâteaux derivative of <i>f</i> at <i>x</i><sub>0</sub> and the limit in (1.1) holds uniformly in u in the unit ball (or unit sphere) in <i>X</i>. An alternative way to state the definition is to require that <p> [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] <p> Thus <i>T</i> defines the natural linear approximation of f in a neighborhood of the point x0. Sometimes <i>T</i> is called the first variation of <i>f</i> at the point <i>x</i><sub>0</sub>. <p> Clearly, for both notions of derivatives we have only to require that <i>f</i> be defined in a neighborhood of <i>x</i><sub>0</sub>. <p> The existence of a derivative of a function <i>f</i> at a point <i>x</i><sub>0</sub> is not obvious. The question of existence of a derivative for functions from R to R was the subject of research and much discussion among mathematicians for a long time, mainly in the nineteenth century. If <i>f</i> : R -> R has a derivative at <i>x</i><sub>0</sub> then it must be continuous at x0. While it is obvious how to construct a continuous function <i>f</i> : R -> R which fails to have a derivative at a given point, the problem of finding such a function which is nowhere differentiable is not easy. The first to do this was the Czech mathematician Bernard Bolzano in an unpublished manuscript about 1820. He did not supply a full proof that his function had indeed the desired properties. Later, around 1850, Bernhard Riemann mentioned in passing such an example. It was found out later that his example was not correct. The first one who published such an example with a valid proof was Karl Weierstrass in 1875. The first general result on existence of derivatives for functions <i>f</i> : R -> R was found by Henri Lebesgue in his thesis (around 1900). He proved that a monotone function <i>f</i> : R -> R is differentiable almost everywhere. As a consequence it follows that every Lipschitz function <i>f</i> : R -> R, that is, a function which satisfies <p> [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] <p> for some constant C and every <i>s; t</i> [member of] R, has a derivative a.e. This result is sharp in the sense that for every <i>A [??] R</i> of measure zero there is a Lipschitz (even monotone) function <i>f</i> : R -> R which fails to have a derivative at any point of <i>A</i>. <p> Lebesgue's result was extended to Lipschitz functions <i>f</i> : R<sup><i>n</i></sup> -> R by Hans Rademacher, who showed that in this case <i>f</i> is also differentiable a.e. However, this result is not as sharp as Lebesgue's: there are planar sets of measure zero that contain points of differentiability of all Lipschitz functions <i>f</i> : R<sup>2</sup> -> R. This can be seen by detailed inspection of our arguments in Chapter 12 (more details are in). Questions related to sharpness of Rademacher's theorem have recently received considerable attention. See, for example. We do not cover this development here since its main interest and deepest results are finite dimensional, whereas our aim is to contribute to the understanding of the infinite dimensional situation. <p> The concept of a Lipschitz function makes sense for functions between metric spaces. Consequently, this gives rise to the study of derivatives of Lipschitz functions between Banach spaces <i>X</i> and <i>Y</i>. It is easy to see that in view of the compactness of balls in finite dimensional Banach spaces both concepts of a derivative, defined above, coincide if dim <i>X</i> < ∞ and <i>f</i> is Lipschitz. However, if dim <i>X</i> = ∞ easy examples show that there is a big difference between Gâteaux and Fréchet differentiability even for simple Lipschitz functions. <p> In the formulation of Lebesgue's theorem there appears the notion of a.e. (almost everywhere). If we consider infinite dimensional spaces and want to extend Lebesgue's theorem to functions on them, we have first to extend the notion of a.e. to such spaces. In other words, we have to define in a reasonable way a family of negligible sets on such spaces. (These sets are also often called exceptional or null.) The negligible sets should form a proper σ-ideal of subsets of the given space <i>X</i>, that is, be closed under subsets and countable unions, and should not contain all subsets of <i>X</i>. Since sets that are involved in differentiability problems are Borel, we can equivalently consider σ-ideals of Borel subsets of <i>X</i>, that is, families of Borel sets, closed under taking Borel subsets and countable unions. It turns out that this can be done in several nonequivalent ways (in our study below we were led to an infinite family of such σ-ideals). Thus the study of derivatives of functions defined on Banach spaces leads in a natural way to questions of descriptive set theory. <p> In the study of differentiation of Lipschitz functions on Banach spaces, one obstacle has been apparent from the outset. It was recognized already in 1930. The isometry <i>t</i> -> 1[0;<i>t</i>] (the indicator function of the interval [0; <i>t</i>]) from the unit interval to <i>L</i><sub>1</sub>[0; 1] does not have a Gâteaux derivative at a single point. The class of Banach spaces where this pathology does not appear was singled out already in the 1930s and characterized in various ways. The "good" Banach spaces (i.e., spaces <sub>X</sub> so that Lipschitz maps from R to X have a derivative a.e.) are now called spaces with the Radon-Nikodym property (or RNP spaces). The reason for this terminology is that one of their characterizations is that a version of the Radon-Nikodym theorem holds for measure with values in them. A detailed study of this class of spaces is presented in the books. All separable conjugate (in particular reflexive) spaces are RNP spaces. As we have just noted, this class does not include Banach spaces having <i>L</i><sub>1</sub>[0; 1] as a subspace. A similar easy argument shows that an RNP space cannot contain c0 as a subspace. More sophisticated arguments are needed to show that there are separable Banach spaces with the RNP which are not subspaces of separable conjugate spaces or that there are spaces which fail to have the RNP but do not contain subspaces isomorphic to L1[0; 1] or c0. Such examples are presented in detail in. <p> The theorem of Lebesgue can be extended to Gâteaux differentiability of Lipschitz functions from an open subset of a separable Banach space into Banach spaces with the RNP. This was done by various authors independently in the 1970s by using different σ-ideals of negligible sets. The proofs are not difficult, and again all the details may be found in. The situation concerning the existence of Gâteaux derivatives is at present quite satisfactory. On the other hand, the question of existence of Fréchet derivatives seems to be deep and our current knowledge concerning it is rather incomplete. This book is devoted to the study of this topic. Most of it consists of new material. We also recall the known results concerning this question and mention several of the problems which are still open. The proofs of most known results in this direction are at present quite difficult. It is not clear to us whether they can be considerably simplified. We present the proofs of the main results with all details and often accompany them with some words of motivation. Some examples we present seem to indicate that the fault in the difficulty lies mainly in the nature of things. <p> In dealing with Fréchet differentiability it turns out quite soon that we have to restrict the Banach spaces that can serve as domain spaces. The function <i>x</i> -> ||x|| from X to the reals is obviously continuous and convex and thus Lipschitz. If <i>X = l</i><sub>1</sub> this function is easily seen not to be Fréchet differentiable at a single point. A similar situation may occur whenever <i>X</i> is separable but <i>X</i><sup>*</sup> is not. A separable Banach space <i>X</i> is called an Asplund space if <i>X</i><sup>*</sup> is again separable. The reason for this terminology is that Asplund was the first to prove that in such spaces real-valued convex and continuous functions have many points of Fréchet differentiability. "Many points" means here a set whose complement is a set of the first category (i.e., small in the sense of category and not in general in the sense of measure). This shows again why the study of Fréchet differentiability is strongly connected to descriptive set theory. Thus our real subject of study in this book is the existence of Fréchet derivatives of Lipschitz functions from <i>X</i> to <i>Y</i> , where <i>X</i> is an Asplund space and <i>Y</i> has the RNP. <p> Perhaps the best known open question about differentiability of Lipschitz mappings is whether every countable collection of real-valued Lipschitz functions on an Asplund space has a common point of Fréchet differentiability. <p> Optimistic conjectures would assert Fréchet differentiability of Lipschitz functions almost everywhere with respect to a suitable proper σ-ideal of exceptional sets (or null sets). Based on what we currently know (including the results proved here), an optimistic differentiability conjecture may be stated in the following way. <p> <p> <b>Conjecture.</b> In every Asplund space <i>X</i> there is a nontrivial notion of exceptional sets such that, for every locally Lipschitz map <i>f</i> of an open subset <i>G [subset] X</i> into a Banach space <i>Y</i> having the RNP: <p> (C1) <i>f</i> is Gâteaux differentiable almost everywhere in <i>G</i>. <p> (C2) If <i>S [subset] G</i> is a set with null complement such that <i>f</i> is Gâteaux differentiable at every point of <i>S</i>, then Lip(<i>f</i>) = sup<i>x[member of]S</i> ||<i>f</i>'(<i>x</i>)||. <p> (C3) If the set of Gâteaux derivatives of <i>f</i> attained on some <i>E [subset] G</i> is norm separable, then <i>f</i> is Fréchet differentiable at almost every point of <i>E</i>. <p> There is very little evidence for validity of this Conjecture in the generality given above. On the positive side it holds if <i>Y = R</i> and, as we shall see in Chapter 6, it also holds for some infinite dimensional spaces <i>X</i>. On the negative side, it is unknown even whether every three real-valued Lipschitz functions on a Hilbert space have a common point of Fréchet differentiability. (The fact that every two real-valued Lipschitz functions on a Hilbert space have a common point of Fréchet differentiability is one of the new results we prove here.) Moreover, as far as we know, the Conjecture fails with any known nontrivial σ-ideal of subsets of infinite dimensional Hilbert spaces. <p> In any detailed study of Fréchet differentiability one immediately encounters the notion of porous sets (and of _-porous sets that are their countable unions). We will give their usual definition later, since at this moment the only relevant fact is that a set <i>E [subset] X</i> is porous if an only if the function <i>x</i> -> dist(<i>x, E</i>) is Fréchet nondifferentiable at any point of <i>E</i>. It follows that porous sets have to belong to the σ-ideal hoped for in the above Conjecture. Of course, it also has to contain the sets of Gâteaux nondifferentiability of Lipschitz maps to RNP spaces. Denoting just for the purpose of this discussion by <i>I</i> the σ-ideal of subset of <i>X</i> generated by the porous sets and sets of Gâteaux nondifferentiability of Lipschitz maps from <i>X</i> into RNP spaces, one may hope that the Conjecture holds with exceptional sets being defined as elements of <i>I</i>. Both subproblems of this variant of the Conjecture are open: it is unknown whether it holds with these exceptional sets, and it is unknown whether <i>I</i> is nontrivial. <p> A weaker variant of the Conjecture than the one at which we arrived above is in fact true. For the λ-null sets (which will be defined in Chapter 5) we show that (C1) and (C2) hold, and that (C3) holds for the given space <i>X</i> if and only if every porous subset in <i>X</i> is λ-null. This result coming from was the first showing that the problem of smallness of porous sets is related to the problem of existence of Fréchet differentiability points of Lipschitz functions. One of the contributions of this text is to bring better understanding of this relation. <p> The statement (C2) is a weak form of the mean value estimate. Although it is stated for vector-valued functions, it can be equivalently asked only for real-valued functions (as the general case follows by considering <i>x</i><sup>*</sup> [??] f for a suitable <i>x</i><sup>*</sup> [member of] <i>X</i><sup>*</sup>). One can argue that without the validity of this statement a differentiability result would not be very useful. For vector-valued functions there is, however, a stronger mean value estimate, the one that one would obtain by estimating in the Gauss-Green divergence theorem the integral of the divergence by its supremum. We will explain this concept, which we call a multidimensional mean value estimate, in detail in the last section of Chapter 2. The main results of this book give a fairly complete answer to the question under what conditions all Lipschitz mapping of <i>X</i> to finite dimensional spaces not only possess points of Fréchet differentiability, but possess so many of them that even the multidimensional mean value estimate holds. It turns out that this property is much stronger that mere existence of points of Fréchet differentiability: for example, for mappings on Hilbert spaces it holds if the target is two-dimensional, but fails if it is three-dimensional. <p> We now describe some of the contents of this book in more detail. Every chapter starts with a brief information about its content and basic relation to results proved elsewhere. In most cases this is followed by an introductory section, which may also state the main results and prove their most important corollaries. However, some chapters contain rather diverse sets of results, in which case their statements are often deferred to the section in which they are proved. The key notions and notation are introduced at the end of this Introduction; more specialized notions and notation are given only when they are needed. The index and index of notation at the end should help the reader to find the definitions quickly. <p> The main point of the starting chapters is to revise some basic notions and results, although they also contain new results or concepts. Proofs that are well covered in the main reference are not repeated here. <p> Chapter 2 recalls the notion of the Radon-Nikodym property and main results on Gâteaux differentiability of Lipschitz functions and related notions of null sets. Throughout the text, we will be interested not only in mere existence of points of Fréchet differentiability, but also, and often more important, in validity of the mean value estimates. We therefore explain this concept here in some detail. In particular, we spend some time on explaining the meaning of multidimensional mean value estimates, as this seems to be the concept behind nearly all positive results as well as the main counterexamples. <p> <i>(Continues...)</i> <p> <p> <!-- copyright notice --> <br></pre> <blockquote><hr noshade size='1'><font size='-2'> Excerpted from <b>Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces</b> by <b>Joram Lindenstrauss David Preiss Jaroslav Tier</b> Copyright © 2012 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.<br>Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.