On Fréchet Differentiability of Lipschitz Maps
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Math 346 Lecture #3 6.3 the General Fréchet Derivative
Math 346 Lecture #3 6.3 The General Fr´echet Derivative We now extend the notion of the Fr´echet derivative to the Banach space setting. Throughout let (X; k · kX ) and (Y; k · kY ) be Banach spaces, and U an open set in X. 6.3.1 The Fr´echet Derivative Definition 6.3.1. A function f : U ! Y is Fr´echet differentiable at x 2 U if there exists A 2 B(X; Y ) such that kf(x + h) − f(x) − A(h)k lim Y = 0; h!0 khkX and we write Df(x) = A. We say f is Fr´echet differentiable on U if f is Fr´echet differ- entiable at each x 2 U. We often refer to Fr´echet differentiable simply as differentiable. Note. When f is Fr´echet differentiable at x 2 U, the derivative Df(x) is unique. This follows from Proposition 6.2.10 (uniqueness of the derivative for finite dimensional X and Y ) whose proof carries over without change to the general Banach space setting (see Remark 6.3.9). Remark 6.3.2. The existence of the Fr´echet derivative does not change when the norm on X is replace by a topologically equivalent one and/or the norm on Y is replaced by a topologically equivalent one. Example 6.3.3. Any L 2 B(X; Y ) is Fr´echet differentiable with DL(x)(v) = L(v) for all x 2 X and all v 2 X. The proof of this is exactly the same as that given in Example 6.2.5 for finite-dimensional Banach spaces. -
Factoring Absolutely Summing Operators Through Hilbert-Schmidt Operators
Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 1989 Factoring absolutely summing operators through Hilbert-Schmidt operators Jarchow, Hans Abstract: Let K be a compact Hausdorff space, and let C(K) be the corresponding Banach spaceof continuous functions on K. It is well-known that every 1-summing operator S:C(K)→l2 is also nuclear, and therefore factors S = S1S2, with S1:l2→l2 a Hilbert-Schmidt operator and S1:C(K)→l2 a bounded operator. It is easily seen that this latter property is preserved when C(K) is replaced by any quotient, and that a Banach space X enjoys this property if and only if its second dual, X**, does. This led A. Pełczyński [15] to ask if the second dual of a Banach space X must be isomorphic to a quotient of a C(K)- space if X has the property that every 1-summing operator X-→l2 factors through a Hilbert-Schmidt operator. In this paper, we shall first of all reformulate the question in an appropriate manner andthen show that counter-examples are available among super-reflexive Tsirelson-like spaces as well as among quasi-reflexive Banach spaces DOI: https://doi.org/10.1017/s0017089500007643 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-154531 Journal Article Published Version Originally published at: Jarchow, Hans (1989). Factoring absolutely summing operators through Hilbert-Schmidt operators. Glasgow Mathematical Journal, 31(02):131-135. DOI: https://doi.org/10.1017/s0017089500007643 FACTORING ABSOLUTELY SUMMING OPERATORS THROUGH HILBERT-SCHMIDT OPERATORS by HANS JARCHOW (Received 10 August, 1987) Introduction. -
How Many Operators Do There Exist on a Banach Space?
HOW MANY OPERATORS DO THERE EXIST ON A BANACH SPACE? TH. SCHLUMPRECHT Abstract. We present some results concerning the following question: Given an infinite dimensional Banach space X. Are there two normalized basic sequences (xn) and (yn), with lim infn!1 kxn − ynk > 0; so that the map xn 7→ yn extends to a linear bounded operator between the linear span (xn) and the linear span (yn)? 1. Introduction Over the last years the following question caught the interest of an increasing number of researchers in the area of infinite dimensional Banach space theory and became for many of them one of the central problems. (Q1) Assume X is an infinite dimensional Banach space. Does there exist a linear bounded operator T on X which is not of the form T = λId + C, where λ is a scalar, C is a compact operator on X, and Id denotes the identity on X? Clearly, if E ⊂ X is a finite dimensional subspace and S : E → X is any linear operator, then S can be extended to a linear operator T : X → X of finite rank. Thus, there must be “many” finite rank operators on X, and thus “many” elements in the closure of the finite rank operators, which is, assuming X has the approximation property, the ideal of compact operators. Of course, the identity must also be a linear bounded operator. Therefore question (Q1) asks wheather or not there are always some other linear bounded operators defined on a Banach space, beyond the class of operators which must exist by elementary reasons. -
FROM CLASSICAL MECHANICS to QUANTUM FIELD THEORY, a TUTORIAL Copyright © 2020 by World Scientific Publishing Co
FROM CLASSICAL MECHANICS TO QUANTUM FIELD THEORY A TUTORIAL 11556_9789811210488_TP.indd 1 29/11/19 2:30 PM This page intentionally left blank FROM CLASSICAL MECHANICS TO QUANTUM FIELD THEORY A TUTORIAL Manuel Asorey Universidad de Zaragoza, Spain Elisa Ercolessi University of Bologna & INFN-Sezione di Bologna, Italy Valter Moretti University of Trento & INFN-TIFPA, Italy World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO 11556_9789811210488_TP.indd 2 29/11/19 2:30 PM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. FROM CLASSICAL MECHANICS TO QUANTUM FIELD THEORY, A TUTORIAL Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-121-048-8 For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11556#t=suppl Desk Editor: Nur Syarfeena Binte Mohd Fauzi Typeset by Stallion Press Email: [email protected] Printed in Singapore Syarfeena - 11556 - From Classical Mechanics.indd 1 02-12-19 3:03:23 PM January 3, 2020 9:10 From Classical Mechanics to Quantum Field Theory 9in x 6in b3742-main page v Preface This book grew out of the mini courses delivered at the Fall Workshop on Geometry and Physics, in Granada, Zaragoza and Madrid. -
Differentiability Problems in Banach Spaces
Derivatives Differentiability problems in Banach spaces For vector valued functions there are two main version of derivatives: Gâteaux (or weak) derivatives and Fréchet (or strong) derivatives. For a function f from a Banach space X 1 David Preiss into a Banach space Y the Gâteaux derivative at a point x0 2 X is by definition a bounded linear operator T : X −! Y so that Expanded notes of a talk based on a nearly finished research monograph “Fréchet differentiability of Lipschitz functions and porous sets in Banach for every u 2 X, spaces” f (x + tu) − f (x ) written jointly with lim 0 0 = Tu t!0 t Joram Lindenstrauss2 and Jaroslav Tišer3 The operator T is called the Fréchet derivative of f at x if it is a with some results coming from a joint work with 0 4 5 Gâteaux derivative of f at x0 and the limit above holds uniformly Giovanni Alberti and Marianna Csörnyei in u in the unit ball (or unit sphere) in X. So T is the Fréchet derivative of f at x0 if 1Warwick 2Jerusalem 3Prague 4Pisa 5London f (x0 + u) = f (x0) + Tu + o(kuk) as kuk ! 0: 1 / 24 2 / 24 Existence of derivatives Sharpness of Lebesgue’s result The first continuous nowhere differentiable f : R ! R was constructed by Bolzano about 1820 (unpublished), who Lebesgue’s result is sharp in the sense that for every A ⊂ of however did not give a full proof. Around 1850, Riemann R measure zero there is a Lipschitz (and monotone) function mentioned such an example, which was later found slightly f : ! which fails to have a derivative at any point of A. -
Value Functions and Optimality Conditions for Nonconvex
Value Functions and Optimality Conditions for Nonconvex Variational Problems with an Infinite Horizon in Banach Spaces∗ H´el`ene Frankowska† CNRS Institut de Math´ematiques de Jussieu – Paris Rive Gauche Sorbonne Universit´e, Campus Pierre et Marie Curie Case 247, 4 Place Jussieu, 75252 Paris, France e-mail: [email protected] Nobusumi Sagara‡ Department of Economics, Hosei University 4342, Aihara, Machida, Tokyo, 194–0298, Japan e-mail: [email protected] February 11, 2020 arXiv:1801.00400v3 [math.OC] 10 Feb 2020 ∗The authors are grateful to the two anonymous referees for their helpful comments and suggestions on the earlier version of this manuscript. †The research of this author was supported by the Air Force Office of Scientific Re- search, USA under award number FA9550-18-1-0254. It also partially benefited from the FJMH Program PGMO and from the support to this program from EDF-THALES- ORANGE-CRITEO under grant PGMO 2018-0047H. ‡The research of this author benefited from the support of JSPS KAKENHI Grant Number JP18K01518 from the Ministry of Education, Culture, Sports, Science and Tech- nology, Japan. Abstract We investigate the value function of an infinite horizon variational problem in the infinite-dimensional setting. Firstly, we provide an upper estimate of its Dini–Hadamard subdifferential in terms of the Clarke subdifferential of the Lipschitz continuous integrand and the Clarke normal cone to the graph of the set-valued mapping describing dynamics. Secondly, we derive a necessary condition for optimality in the form of an adjoint inclusion that grasps a connection between the Euler–Lagrange condition and the maximum principle. -
Gateaux Differentiability Revisited Malek Abbasi, Alexander Kruger, Michel Théra
Gateaux differentiability revisited Malek Abbasi, Alexander Kruger, Michel Théra To cite this version: Malek Abbasi, Alexander Kruger, Michel Théra. Gateaux differentiability revisited. 2020. hal- 02963967 HAL Id: hal-02963967 https://hal.archives-ouvertes.fr/hal-02963967 Preprint submitted on 12 Oct 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Noname manuscript No. (will be inserted by the editor) Gateaux differentiability revisited Malek Abbasi · Alexander Y. Kruger · Michel Thera´ Received: date / Accepted: date Abstract 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 general- ized subdifferential in an arbitrary topological vector space. Meaningful examples preserving the key properties of the original notion of derivative are provided. Keywords Gateaux differentiability · Moreau–Rockafellar subdifferential · Convex function · Directional derivative. Mathematics Subject Classification (2000) 49J52 · 49J53 · 90C30 1 Introduction Gateaux derivatives are widely used in the calculus of variations, optimization and physics. According to Laurent Mazliak [7,8], Gateaux differentiability first appeared in Gateaux’s notes [3,4] under the name variation premiere` . -
ECE 821 Optimal Control and Variational Methods Lecture Notes
ECE 821 Optimal Control and Variational Methods Lecture Notes Prof. Dan Cobb Contents 1 Introduction 3 2 Finite-Dimensional Optimization 5 2.1 Background ........................................ 5 2.1.1 EuclideanSpaces ................................. 5 2.1.2 Norms....................................... 6 2.1.3 MatrixNorms................................... 8 2.2 Unconstrained Optimization in Rn ........................... 9 2.2.1 Extrema...................................... 9 2.2.2 Jacobians ..................................... 9 2.2.3 CriticalPoints................................... 10 2.2.4 Hessians...................................... 11 2.2.5 Definite Matrices . 11 2.2.6 Continuity and Continuous Differentiability . 14 2.2.7 Second Derivative Conditions . 15 2.3 Constrained Optimization in Rn ............................. 16 2.3.1 Constrained Extrema . 16 2.3.2 OpenSets..................................... 17 2.3.3 Strict Inequality Constraints . 18 2.3.4 Equality Constraints and Lagrange Multipliers . 19 2.3.5 Second Derivative Conditions . 21 2.3.6 Non-Strict Inequality Constraints . 23 2.3.7 Mixed Constraints . 27 3 Calculus of Variations 27 3.1 Background ........................................ 27 3.1.1 VectorSpaces ................................... 27 3.1.2 Norms....................................... 28 3.1.3 Functionals .................................... 30 3.2 Unconstrained Optimization in X ............................ 32 3.2.1 Extrema...................................... 32 3.2.2 Differentiation of Functionals . -
The Hitchhiker Guide to Categorical Banach Space Theory. Part I
E extracta mathematicae Vol. 25, N´um.2, 103 { 149 (2010) The Hitchhiker Guide to Categorical Banach Space Theory. Part I. Jesus M. F. Castillo ∗ Departamento de Matem´aticas, Universidad de Extremadura, 06006 Badajoz, Spain [email protected] Received June 1, 2010 Abstract: What has category theory to offer to Banach spacers? In this survey-like paper we will focus on some of the five basic elements of category theory {namely, i) The definition of category, functor and natural transformation; ii) Limits and colimits; iii) Adjoint functors; plus a naive presentation of Kan extensions{ to support the simplest answer \tools that work and a point of view that helps to understand problems, even if one does not care at all about categories". Homology will be treated in a second part. Key words: Categorical Banach space theory, universal constructions, duality and adjoint- ness. AMS Subject Class. (2000): 46Mxx, 18-02, 46-02. 1. Prologue Functional analysis, in general, and Banach space theory, in particular, are unthinkable outside the ambient of algebraic structures. In fact, functional analysis can be defined as the blend of algebra and topology. One of the major achievements of mathematics in the XXth century is category theory, whose paramount importance comes from the fact of establishing a common language for all mathematics. Indeed, it becomes easier for mathematicians working in different areas to communicate if they are able to identify their basic construc- tions, knowing that certain objects of their disciplines are particular instances of the same concept. And the only part of mathematics able today to provide such common language is categorical algebra. -
An Inverse Function Theorem Converse
AN INVERSE FUNCTION THEOREM CONVERSE JIMMIE LAWSON Abstract. We establish the following converse of the well-known inverse function theorem. Let g : U → V and f : V → U be inverse homeomorphisms between open subsets of Banach spaces. If g is differentiable of class Cp and f if locally Lipschitz, then the Fr´echet derivative of g at each point of U is invertible and f must be differentiable of class Cp. Primary 58C20; Secondary 46B07, 46T20, 46G05, 58C25 Key words and phrases. Inverse function theorem, Lipschitz map, Banach space, chain rule 1. Introduction A general form of the well-known inverse function theorem asserts that if g is a differentiable function of class Cp, p ≥ 1, between two open subsets of Banach spaces and if the Fr´echet derivative of g at some point x is invertible, then locally around x, there exists a differentiable inverse map f of g that is also of class Cp. But in various settings, one may have the inverse function f readily at hand and want to know about the invertibility of the Fr´echet derivative of g at x and whether f is of class Cp. Our purpose in this paper is to present a relatively elementary proof of this arXiv:1812.03561v1 [math.FA] 9 Dec 2018 converse result under the general hypothesis that the inverse f is (locally) Lipschitz. Simple examples like g(x)= x3 at x = 0 on the real line show that the assumption of continuity alone is not enough. Thus it is a bit surprising that the mild strengthening to the assumption that the inverse is locally Lipschitz suffices. -
A Generalization of the Implicit Function Theorem
Applied Mathematical Sciences, Vol. 4, 2010, no. 26, 1289 - 1298 A Generalization of the Implicit Function Theorem Elvio Accinelli Facultad de Economia de la UASLP Av. Pintores S/N, Fraccionamiento Burocratas del Estado CP 78263 San Luis Potosi, SLP Mexico [email protected] Abstract In this work, we generalize the classical theorem of the implicit func- tion, for functions whose domains are open subset of Banach spaces in to a Banach space, to the case of functions whose domains are convex subset, not necessarily open, of Banach spaces in to a Banach space. We apply this theorem to show that the excess utility function of an economy with infinitely many commodities, is a differentiable mapping. Keywords: Implicit Function Theorem, convex subsets, Banach spaces 1 Introduction The purpose of this work is to show that the implicit function theorem can be generalized to the case of functions whose domains are defined as a cartesian product of two convex subsets S, and W not necessarily open, of a cartesian product X × Y of Banach spaces, in to a Banach space Z. To prove our main theorem we introduce the concept of Gateaux derivative. However, it is enough the existence of the Gateaux derivatives of a given function, only in admissible directions. Let S ⊂ X be a subset of a Banach space X, and x ∈ S. So, we will say that the vector h ∈ X is admissible for x ∈ S if and only if x + h ∈ S. In order to define the Gateaux derivative of a function in admissible directions in a point x, the only necessary condition is the convexity of the domain of the function. -
DIFFERENTIABILITY PROPERTIES of UTILITY FUNCTIONS Freddy
DIFFERENTIABILITY PROPERTIES OF UTILITY FUNCTIONS Freddy Delbaen Department of Mathematics, ETH Z¨urich Abstract. We investigate differentiability properties of monetary utility functions. At the same time we give a counter-example – important in finance – to automatic continuity for concave functions. Notation and Preliminaries We use standard notation. The triple (Ω, F, P) denotes an atomless probability space. In practice this is not a restriction since the property of being atomless is equivalent to the fact that on (Ω, F, P), we can define a random variable that has a continuous distribution function. We use the usual – unacceptable – convention to identify a random variable with its class (defined by a.s. equality). By Lp we denote the standard spaces, i.e. for 0 <p<∞, X ∈ Lp if and only if E[|X|p] < ∞. L0 stands for the space of all random variables and L∞ is the space of bounded random variables. The topological dual of L∞ is denoted by ba, the space of bounded finitely additive measures µ defined on F with the property that P[A]=0 implies µ(A)=0. The subset of normalised non-negative finitely additive measures – the so called finitely additive probability measures – is denoted by Pba. The subset of countably additive elements of Pba (a subset of L1)isdenoted by P. Definition. A function u: L∞ → R is called a monetary utility function if (1) ξ ∈ L∞ and ξ ≥ 0 implies u(ξ) ≥ 0, (2) u is concave, (3) for a ∈ R and ξ ∈ L∞ we have u(ξ + a)=u(ξ)+a. If u moreover satisfies u(λξ)=λu(ξ) for λ ≥ 0 (positive homogeniety), we say that u is coherent.