Generic Frechet-Differentiability and Perturbed Optimization Problems in Banach Spaces

1976 ◽  
Vol 224 (2) ◽  
pp. 193 ◽  
Author(s):  
Ivar Ekeland ◽  
Gerard Lebourg
Keyword(s):  
Banach Spaces ◽  
Optimization Problems ◽  
Fréchet Differentiability ◽  
Perturbed Optimization ◽  
Frechet Differentiability

Fr ´Echet Differentiability Except For Γ‎-Null Sets

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Spaces ◽  
Common Point ◽  
Lipschitz Functions ◽  
Infinite Dimensional ◽  
Lipschitz Maps ◽  
Almost Everywhere ◽  
Fréchet Differentiability ◽  
Gateaux Derivatives ◽  
Frechet Differentiability ◽  
Porous Sets

This chapter gives an account of the known genuinely infinite dimensional results proving Fréchet differentiability almost everywhere except for Γ‎-null sets. 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 irregularly, and they turn out to be the only obstacle to validity of a Fréchet differentiability result Γ‎-almost everywhere. Furthermore, geometry of the space may (or may not) guarantee that porous sets are Γ‎-null. The chapter also shows that on some infinite dimensional Banach spaces countable collections of real-valued Lipschitz functions, and even of fairly general Lipschitz maps to infinite dimensional spaces, have a common point of Fréchet differentiability.

A Dual characterization of Banach Spaces With the Convex Point-of-Continuity Property

1989 ◽  
Vol 32 (3) ◽  
pp. 274-280
Author(s):  
D. E. G. Hare
Keyword(s):  
Banach Spaces ◽  
Continuity Property ◽  
Dual Spaces ◽  
Equivalent Norms ◽  
Fréchet Differentiability ◽  
New Type ◽  
Frechet Differentiability ◽  
Point Of Continuity Property ◽  
Convex Point

AbstractWe introduce a new type of differentiability, called cofinite Fréchet differentiability. We show that the convex point-of-continuity property of Banach spaces is dual to the cofinite Fréchet differentiability of all equivalent norms. A corresponding result for dual spaces with the weak* convex point-of-continuity property is also established.

Smoothness, Convexity, Porosity, and Separable Determination

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Equivalent Norm ◽  
Fréchet Differentiability ◽  
Smooth Norm ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Set Of Points

This chapter shows how spaces with separable dual admit a Fréchet smooth norm. It first considers a criterion of the differentiability of continuous convex functions on Banach spaces before discussing Fréchet smooth and nonsmooth renormings and Fréchet differentiability of convex functions. It then describes the connection between porous sets and Fréchet differentiability, along with the set of points of Fréchet differentiability of maps between Banach spaces. It also examines the concept of separable determination, the relevance of the σ‎-porous sets for differentiability and proves the existence of a Fréchet smooth equivalent norm on a Banach space with separable dual. The chapter concludes by explaining how one can show that many differentiability type results hold in nonseparable spaces provided they hold in separable ones.

scholarly journals Bifurcation at isolated singular points of the Hadamard derivative

2014 ◽  
Vol 144 (5) ◽  
pp. 1027-1065 ◽  
Author(s):  
C. A. Stuart
Keyword(s):  
Banach Spaces ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Open Neighbourhood ◽  
Lipschitz Continuous ◽  
Nonlinear Elliptic ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Hadamard Derivative ◽  
Shed Light

For Banach spaces X and Y, we consider bifurcation from the line of trivial solutions for the equation F (λ, u) = 0, where F : ℝ × X → Y with F (λ, 0) = 0 for all λ ∈ ℝ. The focus is on the situation where F (λ, ·) is only Hadamard differentiable at 0 and Lipschitz continuous on some open neighbourhood of 0, without requiring any Fréchet differentiability. Applications of the results obtained here to some problems involving nonlinear elliptic equations on ℝN, where Fréchet differentiability is not available, are presented in some related papers, which shed light on the relevance of our hypotheses.

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

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Jaroslav Tier
Keyword(s):  
Banach Spaces ◽  
Set Theory ◽  
Lipschitz Functions ◽  
Descriptive Set Theory ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Descriptive Set ◽  
Vector Valued ◽  
Derivatives Of

This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vector-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysis. The new methods developed here include a game approach to perturbational variational principles that is of independent interest. Detailed explanation of the underlying ideas and motivation behind the proofs of the new results on Fréchet differentiability of vector-valued functions should make these arguments accessible to a wider audience. The most important special case of the differentiability results, that Lipschitz mappings from a Hilbert space into the plane have points of Fréchet differentiability, is given its own chapter with a proof that is independent of much of the work done to prove more general results. The book raises several open questions concerning its two main topics.

ALMOST FRÉCHET DIFFERENTIABILITY OF LIPSCHITZ MAPPINGS BETWEEN INFINITE-DIMENSIONAL BANACH SPACES

2002 ◽  
Vol 84 (3) ◽  
pp. 711-746 ◽  
Author(s):  
WILLIAM B. JOHNSON ◽  
JORAM LINDENSTRAUSS ◽  
DAVID PREISS ◽  
GIDEON SCHECHTMAN
Keyword(s):  
Banach Spaces ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Lipschitz Mapping ◽  
Finite Dimensional Space ◽  
Infinite Dimensional ◽  
Lipschitz Mappings ◽  
Finite Dimensional ◽  
Fréchet Differentiability ◽  
Frechet Differentiability

We give several sufficient conditions on a pair of Banach spaces X and Y under which each Lipschitz mapping from a domain in X to Y has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability. Most of these conditions are stated in terms of the moduli of asymptotic smoothness and convexity, notions which have appeared in the literature under a variety of names. We prove, for example, that for $\infty > r > p \ge 1$, every Lipschitz mapping from a domain in an $\ell_r$-sum of finite-dimensional spaces into an $\ell_p$-sum of finite-dimensional spaces has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability, and that every Lipschitz mapping from an asymptotically uniformly smooth space to a finite-dimensional space has such points. The latter result improves, with a simpler proof, an earlier result of the second and third authors. We also survey some of the known results on the notions of asymptotic smoothness and convexity, prove some new properties, and present some new proofs of existing results.2000 Mathematical Subject Classification: 46G05, 46T20.

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