Real-valued Lipschitz Function Research Articles

Lipschitz-free Banach spaces

We show that when a linear quotient map to a separable Banach space X has a Lipschitz right inverse, then it has a linear right inverse. If a separable space X embeds isometrically into a Banach space Y , then Y contains an isometric linear copy of X. This is false for every nonseparable weakly compactly generated Banach space X. Canonical examples of nonseparable Banach spaces which are Lipschitz isomorphic but not linearly isomorphic are constructed. If a Banach space X has the bounded approximation property and Y is Lipschitz isomorphic to X, then Y has the bounded approximation property. 1. Introduction. A (real) Banach space X is in particular a metric space, equipped with the distinguished point f0g, and to such a pointed metric space we can associate the space Lip 0 (X) of all real-valued Lipschitz functions which vanish at 0. We refer to (29) for basic facts on this space and some of its uses. It is clear that Lip 0 (X) is a Banach space when it is equipped with the Lipschitz norm, dened

Double Operator Integrals

This paper is concerned with perturbation formulae of the form∥f(a)−f(b)∥Lp(M,τ)⩽K∥a−b∥ Lp(M,τ) with K>0 being a constant depending on p and f only, where f is a real-valued Lipschitz function and a,b are self-adjoint τ-measurable operators affiliated with a semifinite von Neumann algebra (M,τ), such that the difference a−b belongs to Lp(M,τ), 1<p<∞. In order to treat the situation where the von Neumann algebra M is not necessarily hyperfinite, we first develop an integration theory with respect to finitely additive spectral measures in a Banach space. Applied to product measures this integration theory may be considered as an abstract version of the double operator integrals due to Birman and Solomyak. To describe the class of integrable functions we employ our recent study of multiplier theory in UMD-spaces. Our perturbation formulae extend those of Davies and Birman–Solomyak for the case when M is a hyperfinite I∞-factor (i.e., for the Schatten p-classes). We also discuss analogous perturbation results in the setting of symmetric operator spaces associated with (M,τ).

A new proof of Fréchet differentiability of Lipschitz functions

Abstract. We give a relatively simple (self-contained) proof that every real-valued Lipschitz function on Ê2 (or more generally on an Asplund space) has points of Fréchet differentiability. Somewhat more generally, we show that a real-valued Lipschitz function on a separable Banach space has points of Fréchet differentiability provided that the w* closure of the set of its points of Gâteaux differentiability is norm separable.

TYPICAL PROPERTIES OF LIPSCHITZ FUNCTIONS

In terms of Baire category, a typical real-valued Lipschitz function on a finite dimensional space has a local minimum at every point of a dense subset of the domain, and a Dini subdifferential that is either singleton or empty at all points. Moreover, its Dini subdifferential is empty outside a set of first category. Hence a typical Lipschitz function has no points of subdifferential regularity.

Distinct differentiable functions may share the same Clarke subdifferential at all points

We construct, using Zahorski’s Theorem, two everywhere differentiable real–valued Lipschitz functions differing by more than a constant but sharing the same Clarke subdifferential and the same approximate subdifferential.

Approximate subgradients and coderivatives in R n

We show that in two dimensions or higher, the Mordukhovich-Ioffe approximate subdifferential and Clarke subdifferential may differ almost everywhere for real-valued Lipschitz functions. Uncountably many Frechet differentiable vector-valued Lipschitz functions differing by more than constants can share the same Mordukhovich-Ioffe coderivatives. Moreover, the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be nonconvex almost everywhere for Frechet differentiable vector-valued Lipschitz functions. Finally we show that for vector-valued Lipschitz functions the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be almost everywhere disconnected.

Characterization of nonsmooth functions through their generalized gradients

Given a real-valued Lipschitz function, we explore properties of its generalized gradient in Clarke's sense that corresponds to f being quasi-convex, to f being d.c, (difference of convex functions), to f being a pointwise supremum of functions that are ktimes continuously differentiable. In other words, knowing that ∂f is the generalized gradient multifunction of a function f what kind of properties of ∂f could serve to characterize f?. The classes of nonsmooth function involed in this paper are: quasi-function, d.e functions, lower-C k functions, semi smooth functions and quasidifferentiable functions.

Gateaux differentiable functions are somewhere Frechet differentiable

It is proved that every everywhere Gateaux differentiable real-valued Lipschitz function on an Asplund space is Frechet differentiable at uncountably many points. The nonseparable case is reduced to the separable one with the help of a general separable reduction theorem.