An Inverse Function Theorem Converse
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Inverse Vs Implicit Function Theorems - MATH 402/502 - Spring 2015 April 24, 2015 Instructor: C
Inverse vs Implicit function theorems - MATH 402/502 - Spring 2015 April 24, 2015 Instructor: C. Pereyra Prof. Blair stated and proved the Inverse Function Theorem for you on Tuesday April 21st. On Thursday April 23rd, my task was to state the Implicit Function Theorem and deduce it from the Inverse Function Theorem. I left my notes at home precisely when I needed them most. This note will complement my lecture. As it turns out these two theorems are equivalent in the sense that one could have chosen to prove the Implicit Function Theorem and deduce the Inverse Function Theorem from it. I showed you how to do that and I gave you some ideas how to do it the other way around. Inverse Funtion Theorem The inverse function theorem gives conditions on a differentiable function so that locally near a base point we can guarantee the existence of an inverse function that is differentiable at the image of the base point, furthermore we have a formula for this derivative: the derivative of the function at the image of the base point is the reciprocal of the derivative of the function at the base point. (See Tao's Section 6.7.) Theorem 0.1 (Inverse Funtion Theorem). Let E be an open subset of Rn, and let n f : E ! R be a continuously differentiable function on E. Assume x0 2 E (the 0 n n base point) and f (x0): R ! R is invertible. Then there exists an open set U ⊂ E n containing x0, and an open set V ⊂ R containing f(x0) (the image of the base point), such that f is a bijection from U to V . -
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. -
Quantum Query Complexity and Distributed Computing ILLC Dissertation Series DS-2004-01
Quantum Query Complexity and Distributed Computing ILLC Dissertation Series DS-2004-01 For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam phone: +31-20-525 6051 fax: +31-20-525 5206 e-mail: [email protected] homepage: http://www.illc.uva.nl/ Quantum Query Complexity and Distributed Computing Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof.mr. P.F. van der Heijden ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Aula der Universiteit op dinsdag 27 januari 2004, te 12.00 uur door Hein Philipp R¨ohrig geboren te Frankfurt am Main, Duitsland. Promotores: Prof.dr. H.M. Buhrman Prof.dr.ir. P.M.B. Vit´anyi Overige leden: Prof.dr. R.H. Dijkgraaf Prof.dr. L. Fortnow Prof.dr. R.D. Gill Dr. S. Massar Dr. L. Torenvliet Dr. R.M. de Wolf Faculteit der Natuurwetenschappen, Wiskunde en Informatica The investigations were supported by the Netherlands Organization for Sci- entific Research (NWO) project “Quantum Computing” (project number 612.15.001), by the EU fifth framework projects QAIP, IST-1999-11234, and RESQ, IST-2001-37559, the NoE QUIPROCONE, IST-1999-29064, and the ESF QiT Programme. Copyright c 2003 by Hein P. R¨ohrig Revision 411 ISBN: 3–933966–04–3 v Contents Acknowledgments xi Publications xiii 1 Introduction 1 1.1 Computation is physical . 1 1.2 Quantum mechanics . 2 1.2.1 States . -
1. Inverse Function Theorem for Holomorphic Functions the Field Of
1. Inverse Function Theorem for Holomorphic Functions 2 The field of complex numbers C can be identified with R as a two dimensional real vector space via x + iy 7! (x; y). On C; we define an inner product hz; wi = Re(zw): With respect to the the norm induced from the inner product, C becomes a two dimensional real Hilbert space. Let C1(U) be the space of all complex valued smooth functions on an open subset U of ∼ 2 C = R : Since x = (z + z)=2 and y = (z − z)=2i; a smooth complex valued function f(x; y) on U can be considered as a function F (z; z) z + z z − z F (z; z) = f ; : 2 2i For convince, we denote f(x; y) by f(z; z): We define two partial differential operators @ @ ; : C1(U) ! C1(U) @z @z by @f 1 @f @f @f 1 @f @f = − i ; = + i : @z 2 @x @y @z 2 @x @y A smooth function f 2 C1(U) is said to be holomorphic on U if @f = 0 on U: @z In this case, we denote f(z; z) by f(z) and @f=@z by f 0(z): A function f is said to be holomorphic at a point p 2 C if f is holomorphic defined in an open neighborhood of p: For open subsets U and V in C; a function f : U ! V is biholomorphic if f is a bijection from U onto V and both f and f −1 are holomorphic. A holomorphic function f on an open subset U of C can be identified with a smooth 2 2 mapping f : U ⊂ R ! R via f(x; y) = (u(x; y); v(x; y)) where u; v are real valued smooth functions on U obeying the Cauchy-Riemann equation ux = vy and uy = −vx on U: 2 2 For each p 2 U; the matrix representation of the derivative dfp : R ! R with respect to 2 the standard basis of R is given by ux(p) uy(p) dfp = : vx(p) vy(p) In this case, the Jacobian of f at p is given by 2 2 0 2 J(f)(p) = det dfp = ux(p)vy(p) − uy(p)vx(p) = ux(p) + vx(p) = jf (p)j : Theorem 1.1. -
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. -
Inverse and Implicit Function Theorems for Noncommutative
Inverse and Implicit Function Theorems for Noncommutative Functions on Operator Domains Mark E. Mancuso Abstract Classically, a noncommutative function is defined on a graded domain of tuples of square matrices. In this note, we introduce a notion of a noncommutative function defined on a domain Ω ⊂ B(H)d, where H is an infinite dimensional Hilbert space. Inverse and implicit function theorems in this setting are established. When these operatorial noncommutative functions are suitably continuous in the strong operator topology, a noncommutative dilation-theoretic construction is used to show that the assumptions on their derivatives may be relaxed from boundedness below to injectivity. Keywords: Noncommutive functions, operator noncommutative functions, free anal- ysis, inverse and implicit function theorems, strong operator topology, dilation theory. MSC (2010): Primary 46L52; Secondary 47A56, 47J07. INTRODUCTION Polynomials in d noncommuting indeterminates can naturally be evaluated on d-tuples of square matrices of any size. The resulting function is graded (tuples of n × n ma- trices are mapped to n × n matrices) and preserves direct sums and similarities. Along with polynomials, noncommutative rational functions and power series, the convergence of which has been studied for example in [9], [14], [15], serve as prototypical examples of a more general class of functions called noncommutative functions. The theory of non- commutative functions finds its origin in the 1973 work of J. L. Taylor [17], who studied arXiv:1804.01040v2 [math.FA] 7 Aug 2019 the functional calculus of noncommuting operators. Roughly speaking, noncommutative functions are to polynomials in noncommuting variables as holomorphic functions from complex analysis are to polynomials in commuting variables. -
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. -
On Fréchet Differentiability of Lipschitz Maps
Annals of Mathematics, 157 (2003), 257–288 On Fr´echet differentiability of Lipschitz maps between Banach spaces By Joram Lindenstrauss and David Preiss Abstract Awell-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point ofFr´echet differentiability. We show that the answer is positive for some infinite-dimensional X. Previously, even for collections consisting of two functions this has been known for finite-dimensional X only (although for one function the answer is known to be affirmative in full generality). Our aims are achieved by introducing a new class of null sets in Banach spaces (called Γ-null sets), whose definition involves both the notions of category and mea- sure, and showing that the required differentiability holds almost everywhere with respect to it. We even obtain existence of Fr´echet derivatives of Lipschitz functions between certain infinite-dimensional Banach spaces; no such results have been known previously. Our main result states that a Lipschitz map between separable Banach spaces is Fr´echet differentiable Γ-almost everywhere provided that it is reg- ularly Gˆateaux differentiable Γ-almost everywhere and the Gˆateaux deriva- tives stay within a norm separable space of operators. It is easy to see that Lipschitz maps of X to spaces with the Radon-Nikod´ym property are Gˆateaux differentiable Γ-almost everywhere. Moreover, Gˆateaux differentiability im- plies regular Gˆateaux differentiability with exception of another kind of neg- ligible sets, so-called σ-porous sets. The answer to the question is therefore positive in every space in which every σ-porous set is Γ-null. -
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` . -
The Implicit Function Theorem and Free Algebraic Sets Jim Agler
Washington University in St. Louis Washington University Open Scholarship Mathematics Faculty Publications Mathematics and Statistics 5-2016 The implicit function theorem and free algebraic sets Jim Agler John E. McCarthy Washington University in St Louis, [email protected] Follow this and additional works at: https://openscholarship.wustl.edu/math_facpubs Part of the Algebraic Geometry Commons Recommended Citation Agler, Jim and McCarthy, John E., "The implicit function theorem and free algebraic sets" (2016). Mathematics Faculty Publications. 25. https://openscholarship.wustl.edu/math_facpubs/25 This Article is brought to you for free and open access by the Mathematics and Statistics at Washington University Open Scholarship. It has been accepted for inclusion in Mathematics Faculty Publications by an authorized administrator of Washington University Open Scholarship. For more information, please contact [email protected]. The implicit function theorem and free algebraic sets ∗ Jim Agler y John E. McCarthy z U.C. San Diego Washington University La Jolla, CA 92093 St. Louis, MO 63130 February 19, 2014 Abstract: We prove an implicit function theorem for non-commutative functions. We use this to show that if p(X; Y ) is a generic non-commuting polynomial in two variables, and X is a generic matrix, then all solutions Y of p(X; Y ) = 0 will commute with X. 1 Introduction A free polynomial, or nc polynomial (nc stands for non-commutative), is a polynomial in non-commuting variables. Let Pd denote the algebra of free polynomials in d variables. If p 2 Pd, it makes sense to think of p as a function that can be evaluated on matrices. -
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 .