Differentiability of Lipschitz Maps on Hilbert Spaces

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Lipschitz Maps ◽  
Lipschitz Map ◽  
Fréchet Differentiability ◽  
Dimensional Version ◽  
Frechet Differentiability ◽  
Irregular Behavior

This chapter presents a separate, essentially self-contained, nonvariational proof of existence of points of Fréchet differentiability of R²-valued Lipschitz maps on Hilbert spaces. It begins with the theorem stating that every Lipschitz map of a Hilbert space to a two-dimensional space has points of Fréchet differentiability. This is followed by a lemma, which is stated in an arbitrary Hilbert space but whose validity in the general case follows from its three-dimensional version. The chapter then explains the proof of the theorem and of the lemma stated above. In particular, it considers two cases, one corresponding to irregular behavior and the other to regular behavior.

Asymptotic Fr echet ´Differentiability

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Hilbert Spaces ◽  
Variational Approach ◽  
Three Dimensional ◽  
Mean Value ◽  
Two Dimensional ◽  
Current Development ◽  
Lipschitz Mappings ◽  
Higher Dimensional ◽  
Fréchet Differentiability ◽  
Frechet Differentiability

This chapter presents the current development of the first, unpublished proof of existence of points Fréchet differentiability of Lipschitz mappings to two-dimensional spaces. For functions into higher dimensional spaces the method does not lead to a point of Gâteaux differentiability but constructs points of asymptotic Fréchet differentiability. The proof uses perturbations that are not additive, rather than the variational approach, but still provides (asymptotic) Fréchet derivatives in every slice of Gâteaux derivatives. However, it cannot be used to prove existence of points of Fréchet differentiability of Lipschitz mappings of Hilbert spaces to three-dimensional spaces. The results are negative in the sense that an appropriate version of the multidimensional mean value estimate holds.

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.

scholarly journals Lipschitz Extensions to Finitely Many Points

2018 ◽  
Vol 6 (1) ◽  
pp. 174-191 ◽  
Author(s):  
Giuliano Basso
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Metric Spaces ◽  
Lipschitz Constant ◽  
Target Space ◽  
Square Root ◽  
Lipschitz Maps ◽  
Lipschitz Map ◽  
Lipschitz Extensions

AbstractWe consider Lipschitz maps with values in quasi-metric spaces and extend such maps to finitely many points. We prove that in this context every 1-Lipschitz map admits an extension such that its Lipschitz constant is bounded from above by the number of added points plus one. Moreover, we prove that if the source space is a Hilbert space and the target space is a Banach space, then there exists an extension such that its Lipschitz constant is bounded from above by the square root of the total of added points plus one. We discuss applications to metric transforms.

Porosity and ε‎-Fr échet differentiability

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Porous Set ◽  
Delta Set ◽  
Lipschitz Maps ◽  
Finite Dimensional ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Null Set

This chapter demonstrates that the results about smallness of porous sets, and so also of sets of irregularity points of a given Lipschitz function, can be used to show existence of points of (at least) ε‎-Fréchet differentiability of vector-valued functions. The approach involves combining this new idea with the basic notion that points of ε‎-Fréchet differentiability should appear in small slices of the set of Gâteaux derivatives. The chapter obtains very precise results on existence of points of ε‎-Fréchet differentiability for Lipschitz maps with finite dimensional range. The main result applies when every porous set is contained in the unions of a σ‎-directionally porous (and hence Haar null) set and a Γ‎ₙ-null Gsubscript Small Delta set.

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.

scholarly journals Space and Time as a Consequence of Ghirardi-Rimini-Weber Quantum Jumps

2018 ◽  
Vol 73 (10) ◽  
pp. 923-929 ◽  
Author(s):  
Tejinder P. Singh
Keyword(s):  
Hilbert Space ◽  
Wave Function ◽  
Measurement Problem ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Physical Space ◽  
Space Time ◽  
Time Operator ◽  
Quantum Jumps ◽  
Macroscopic Objects

AbstractThe Ghirardi-Rimini-Weber theory of spontaneous collapse offers a possible resolution of the quantum measurement problem. In this theory, the wave function of a particle spontaneously and repeatedly localises to one or the other random position in space, as a consequence of the hypothesised quantum jumps. In between jumps, the wave function undergoes the usual Schrödinger evolution. In the present paper, we suggest that these jumps take place in Hilbert space, with no reference to physical space and a physical three-dimensional space arises as a consequence of localisation of macroscopic objects in the universe. That is, collapse of the wave-function is responsible for the origin of space. We then suggest that similar jumps take place for a hypothetical time operator in Hilbert space and classical time, as we know it emerges from localisation of this time operator, for macroscopic objects. More generally, the jumps are suggested to take place in an operator space-time in Hilbert space, leading to an emergent classical space-time.

Fr ´Echet Differentiability of Vector-Valued Functions

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Mean Value ◽  
Common Point ◽  
Lipschitz Functions ◽  
Lipschitz Map ◽  
Fréchet Differentiability ◽  
Smooth Norm ◽  
Frechet Differentiability ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Special Case

This chapter shows that if a Banach space with a Fréchet smooth norm is asymptotically smooth with modulus o(tⁿ logⁿ⁻¹(1/t)) then every Lipschitz map of X to a space of dimension not exceeding n has many points of Fréchet differentiability. In particular, it proves that two real-valued Lipschitz functions on a Hilbert space have a common point of Fréchet differentiability. The chapter first presents the theorem whose assumptions hold for any space X with separable dual, includes the result that real-valued Lipschitz functions on such spaces have points of Fréchet differentiability, and takes into account the corresponding mean value estimate. The chapter then gives the estimate for a “regularity parameter” and reduces the theorem to a special case. Finally, it discusses simplifications of the arguments of the proof of the main result in some special situations.

Preliminaries to Main Results

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Flat Surface ◽  
Tensor Products ◽  
Linear Operators ◽  
Euclidean Spaces ◽  
Lipschitz Maps ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Fréchet Differentiable ◽  
Derivatives Of ◽  
Surface Passing

This chapter presents a number of results and notions that will be used in subsequent chapters. In particular, it considers the concept of regular differentiability and the lemma on deformation of n-dimensional surfaces. The idea is to deform a flat surface passing through a point x (along which we imagine that a certain function f is almost affine) to a surface passing through a point witnessing that f is not Fréchet differentiable at x. This is done in such a way that certain “energy” associated to surfaces increases less than the “energy” of the functionf along the surface. The chapter also discusses linear operators and tensor products, various notions and notation related to Fréchet differentiability, and deformation of surfaces controlled by ω‎ⁿ. Finally, it proves some integral estimates of derivatives of Lipschitz maps between Euclidean spaces (not necessarily of the same dimension).

Unavoidable Porous Sets and Nondifferentiable Maps

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Space ◽  
Mean Value ◽  
Porous Set ◽  
Lipschitz Maps ◽  
Fréchet Differentiability ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Frechet Differentiability ◽  
Porous Sets ◽  
Derivatives Of

This chapter discusses Γ‎ₙ-nullness of sets porous “¹at infinity” and/or existence of many points of Fréchet differentiability of Lipschitz maps into n-dimensional spaces. The results reveal a σ‎-porous set whose complement is null on all n-dimensional surfaces and the multidimensional mean value estimates fail even for ε‎-Fréchet derivatives. Previous chapters have established conditions on a Banach space X under which porous sets in X are Γ‎ₙ-null and/or the the multidimensional mean value estimates for Fréchet derivatives of Lipschitz maps into n-dimensional spaces hold. This chapter investigates in what sense the assumptions of these main results are close to being optimal.

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