Über die Autorin bzw. den Autor

Joram Lindenstrauss is professor emeritus of mathematics at the Hebrew University of Jerusalem. David Preiss is professor of mathematics at the University of Warwick. Jaroslav Tier is associate professor of mathematics at Czech Technical University in Prague.

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Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces

PRINCETON UNIVERSITY PRESS

Contents

1 Introduction.......................................................................12 Gâteaux differentiability of Lipschitz functions..............................123 Smoothness, convexity, porosity, and separable determination.......................234 e"-Fréchet differentiability..........................................465 ?-null and ?n-null sets...........................726 Fréchet differentiability except for ?-null sets.......................967 Variational principles.............................................................1208 Smoothness and asymptotic smoothness...............................................1339 Preliminaries to main results......................................................15610 Porosity, ?n -and lambda]-null sets.....................16911 Porosity and e-Fréchet differentiability.............................20212 Fréchet differentiability of real-valued functions...........................22213 Fréchet differentiability of vector-valued functions.........................26214 Unavoidable porous sets and nondifferentiable maps................................31915 Asymptotic Fréchet differentiability.........................................35516 Differentiability of Lipschitz maps on Hilbert spaces.............................392Bibliography.........................................................................415Index................................................................................419Index of Notation....................................................................423

Chapter One

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 f from a Banach space X into a Banach space Y the Gâteaux derivative at a point x₀ [member of] X is by definition a bounded linear operator T : X -> Y such that for every u [member of] X,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

The operator T is called the Fréchet derivative of f at x₀ if it is a Gâteaux derivative of f at x₀ and the limit in (1.1) holds uniformly in u in the unit ball (or unit sphere) in X. An alternative way to state the definition is to require that

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Thus T defines the natural linear approximation of f in a neighborhood of the point x0. Sometimes T is called the first variation of f at the point x₀.

Clearly, for both notions of derivatives we have only to require that f be defined in a neighborhood of x₀.

The existence of a derivative of a function f at a point x₀ 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 f : R -> R has a derivative at x₀ then it must be continuous at x0. While it is obvious how to construct a continuous function f : 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 f : R -> R was found by Henri Lebesgue in his thesis (around 1900). He proved that a monotone function f : R -> R is differentiable almost everywhere. As a consequence it follows that every Lipschitz function f : R -> R, that is, a function which satisfies

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for some constant C and every s; t [member of] R, has a derivative a.e. This result is sharp in the sense that for every A [??] R of measure zero there is a Lipschitz (even monotone) function f : R -> R which fails to have a derivative at any point of A.

Lebesgue's result was extended to Lipschitz functions f : Rⁿ -> R by Hans Rademacher, who showed that in this case f 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 f : R² -> 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.

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 X and Y. 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 X < 8 and f is Lipschitz. However, if dim X = 8 easy examples show that there is a big difference between Gâteaux and Fréchet differentiability even for simple Lipschitz functions.

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 s-ideal of subsets of the given space X, that is, be closed under subsets and countable unions, and should not contain all subsets of X. Since sets that are involved in differentiability problems are Borel, we can equivalently consider s-ideals of Borel subsets of X, 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 s-ideals). Thus the study of derivatives of functions defined on Banach spaces leads in a natural way to questions of descriptive set theory.

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 t -> 1[0;t] (the indicator function of the interval [0; t]) from the unit interval to...