Construction of pathological Gâteaux differentiable functions (original) (raw)

Applied Mathematics & Optimization, 2021

We revisit some basic concepts and ideas of the classical differential calculus and convex analysis extending them to a broader frame. We reformulate and generalize the notion of Gateaux differentiability and propose new notions of generalized derivative and generalized subdifferential in an arbitrary topological vector space. Meaningful examples preserving the key properties of the original notion of derivative are provided.

Generic Gateaux differentiability via smooth perturbations

Bulletin of the Australian Mathematical Society, 1997

We prove that in a Banach space with a Lipschitz uniformly Gateaux smooth bump function, every continuous function which is directionally differentiable on a dense Gj subset of the space, is Gateaux differentiable on a dense Gs subset of the space. Applications of this result are given. The usual applications of variational principles in Banach spaces are to differentiability of real valued functions. For example, the papers [1] and [2] contain results about Gateaux differentiability on dense sets. An application of Ekeland's variational principle to generic Frechet differentiability is given in the proof of the famous Ekeland-Lebourg theorem (see [4]). In [6] an application of the smooth variational principle to generic Gateaux differentiability is presented. In this paper we prove some results about generic Gateaux differentiability of directionally differentiable functions. The tool for proving the main result (Theorem 2) is Proposition 1, which localises precisely the d-minimum point of the perturbed function. The estimate of this localisation is the same as in the Ekeland variational principle. Denote by Lj the Lipschitz constant of a Lipschitz function / : E-» R and by 5, B[x;r] (respectively B(x;r))-the unit sphere of E and the closed (respectively open) ball with center x and radius r. A function b : E-> R is said to be a bump function, if there exists a bounded subset supp b, such that b(x)-0 for every x 0 supp b.

On the range of the derivative of G�teaux-smooth functions on separable banach spaces

Isr J Math, 2005

We prove that there exists a Lipschitz function from ℓ 1 into IR 2 which is Gâteaux-differentiable at every point and such that for every x, y ∈ ℓ 1 , the norm of f ′ (x) − f ′ (y) is bigger than 1. On the other hand, for every Lipschitz and Gâteaux-differentiable function from an arbitrary Banach space X into IR and for every ε > 0, there always exists two points x, y ∈ X such that f ′ (x)−f ′ (y) is less than ε. We also construct, in every infinite dimensional separable Banach space, a real valued function f on X, which is Gâteaux-differentiable at every point, has bounded non-empty support, and with the properties that f ′ is norm to weak * continuous and f ′ (X) has an isolated point a, and that necessarily a = 0.

Geometry and Gâteaux smoothness in separable Banach spaces

Operators and Matrices, 2012

It is a classical fact, due to Day, that every separable Banach space admits an equivalent Gâteaux smooth renorming. In fact, it admits an equivalent uniformly Gâteaux smooth norm, as was shown later byŠmulyan. It is therefore rather unexpected that the existence of Gâteaux smooth renormings satisfying various quantitative estimates on the directional derivative has rather strong structural and geometrical implications for the space. For example, by a result of Vanderwerff, if the directional derivatives satisfy a p-estimate, where p varies arbitrarily with respect to the point and the direction in question, then the Banach space must be an Asplund space. In the present survey paper, we discuss the interplay between various types of Gâteaux differentiability of norms and extreme points with the geometry of separable Banach spaces. In particular, we present various characterizations of Asplund, reflexive, superreflexive, and other classes of separable Banach spaces, via smooth as well as rotund renormings. We also include open problems of various levels of difficulty, with the hope of stimulating research in the area of smoothness and renormings of Banach spaces. In nonlinear analysis, the differentiability of norms plays an important role. The most important type of differentiability is that of Fréchet differentiability. However, in many instances it suffices to use weaker forms of differentiability, i.e., variants of the Gâteaux differentiability (that are more often accessible). This happens especially when some convexity arguments can be combined with Baire category techniques. The present paper surveys some of these results and discusses several ideas and constructions in their proofs. We focus on the interplay of these concepts with the geometry of separable spaces, for example with problems on containment of c 0 or 1 , with superreflexivity, the Radon-Nikodým property, etc. Several open problems in this area are discussed. We refer to, e.g., [Gode], [DGZb], [Fab], [AlKal06], [BoVa10], and [FHHMZ] for all unexplained notions and results used in this note.

On the range of the derivative of a smooth function and applications

2006

We survey recent results on the structure of the range of the derivative of a smooth real valued function f defined on a real Banach space X and of a smooth mapping F between two real Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of L(X, Y) for the existence of a Fréchet-differentiable mapping F from X into Y so that F (X) = A. Whenever F is only assumed Gâteaux-differentiable, new phenomena appear : we discuss the existence of a mapping F from a Banach space X into a Banach space Y , which is bounded, Lipschitz-continuous, and so that for all x, y ∈ X, if x = y, then F (x) − F (y) L(X,Y) > 1. Applications are given to existence and uniqueness of solutions of Hamilton-Jacobi equations. ESTRUCTURA DEL RANGO DE LA DERIVADA DE UNA FUNCIÓN Resumen. Recogemos recientes resultados sobre la estructura del rango de la derivada de una función real f definida en un espacio de Banach real X y de una aplicación diferenciable F entre dos espacios de Banach reales X e Y. Listamos algunas condiciones necesarias y otras suficientes acerca de un subconjunto A de L(X, Y) para la existencia de una aplicación diferenciable Fréchet F de X en Y de modo que F (X) = A. Cuando se supone tan solo que F es Gâteaux diferenciable, aparecen nuevos fenómenos: discutimos la existencia de una aplicación F de un espacio de Banach X en un espacio de Banach Y acotada, Lipschitz-continua, de tal manera que, para todo x, y ∈ X, si x = y, entonces F (x) − F (y) L(X,Y) > 1. Se dan aplicaciones a la existencia de soluciones de ecuaciones de Hamilton-Jacobi.

On the range of the derivative of a smooth mapping between Banach spaces

Abstract and Applied Analysis, 2005

We survey recent results on the structure of the range of the derivative of a smooth mapping f between two Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of ᏸ(X,Y) for the existence of a Fréchet differentiable mapping f from X into Y so that f (X) = A. Whenever f is only assumed Gâteaux differentiable, new phenomena appear: for instance, there exists a mapping f from 1 (N) into R 2 , which is bounded, Lipschitz-continuous, and so that for all x, y ∈ 1 (N), if x = y, then f (x) − f (y) > 1.

Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces

Bulletin of the Australian Mathematical Society, 1993

For a locally Lipschitz function on a separable Banach space the set of points of Gateaux differentiability is dense but not necessarily residual. However, the set of points where the upper Dini derivative and the Clarke derivative agree is residual. It follows immediately that the set of points of intermediate differentiability is also residual and the set of points where the function is Gateaux but not strictly differentiate is of the first category.

Locally Lipschitz Functions and Bornological Derivatives

1993

We study the relationships between Gateaux, weak Hadamard and Frechet differentiability and their bornologies for Lipschitz and for convex functions. In particular, Frechet and weak Hadamard differentiabily coincide for all Lipschitz functions if and only if the space is reflexive (an earlier paper of the first two authors shows that these two notions of differentiability coincide for continuous convex functions

On the range of the derivatives of a smooth function between Banach spaces

Mathematical Proceedings of the Cambridge Philosophical Society, 2003

We study the size of the range of the derivatives of a smooth function between Banach spaces. We establish conditions on a pair of Banach spaces X and Y to ensure the existence of a C p smooth (Fréchet smooth or a continuous Gâteaux smooth) function f from X onto Y such that f vanishes outside a bounded set and all the derivatives of f are surjections. In particular we deduce the following results. For the Gâteaux case, when X and Y are separable and X is infinite-dimensional, there exists a continuous Gâteaux smooth function f from X to Y , with bounded support, so that f (X) = L(X, Y). In the Fréchet case, we get that if a Banach space X has a Fréchet smooth bump and dens X = dens L(X, Y), then there is a Fréchet smooth function f : X −→ Y with bounded support so that f (X) = L(X, Y). Moreover, we see that if X has a C p smooth bump with bounded derivatives and dens X = dens L m s (X; Y) then there exists another C p smooth function f : X −→ Y so that f (k) (X) = L k s (X; Y) for all k = 0, 1, ..., m. As an application, we show that every bounded starlike body on a separable Banach space X with a (Fréchet or Gâteaux) smooth bump can be uniformly approximated by smooth bounded starlike bodies whose cones of tangent hyperplanes fill the dual space X *. In the non-separable case, we prove that X has such property if X has smooth partitions of unity.

The Gâteaux derivative and orthogonality in C∞

Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2012

The general problem in this paper is minimizing the C ∞ − norm of suitable affine mappings from B(H) to C ∞ , using convex and differential analysis (Gateaux derivative) as well as input from operator theory. The mappings considered generalize the so-called elementary operators and in particular the generalized derivations, which are of great interest by themselves. The main results obtained characterize global minima in terms of (Banach space) orthogonality.

The smooth variational principle and generic differentiability

Bulletin of the Australian Mathematical Society, 1991

A modified version of the smooth variational principle of Borwein and Preiss is proved. By its help it is shown that in a Banach space with uniformly Gâteaux differentiable norm every continuous function, which is directionally differentiable on a dense Gδ subset of the space, is Gâteaux differentiable on a dense Gδ subset of the space.

Differentiability and Norming Subspaces

Descriptive Topology and Functional Analysis II, 2019

This is a survey around a property (Property P) introduced by M. Fabian, V. Zizler, and the third named author, in terms of differentiability of the norm. Precisely, a Banach space X is said to have property P if for every norming subspace N ⊂ X * there exists an equivalent Gâteaux differentiable norm for which N is 1-norming. Every weakly compactly generated space has property P. Applications to measure theory, the classification of compacta, and some other structural properties of compact and Banach spaces are given. Some open problems are listed, too. It is based on an earlier paper by Fabian, Zizler, and the third named author, and a recent one by the authors of the survey.