DOCSLIB.ORG

Best Approximations and Orthogonality in Hilbert C*-Modules

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Best Approximations and Orthogonality in Hilbert C*-Modules Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Best approximations and orthogonality in Hilbert C ∗-modules Sushil Singla Department of Mathematics School of Natural Sciences Shiv Nadar University May 11, 2021 1 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Table of Contents 1 Birkhoff-James orthogonality in Hilbert C ∗-modules 2 Proofs of characterization of orthogonality in C ∗-algebra 2 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Table of Contents 1 Birkhoff-James orthogonality in Hilbert C ∗-modules 2 Proofs of characterization of orthogonality in C ∗-algebra 3 / 23 If we can define some kind of orthogonality in a normed space, then we might be able to guess some results from geometric intuition and then try to prove it. Let us see a characterization of a vector being orthogonal to a subspace in an inner product space. Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Idea of Birkhoff-James Orthogonality Question: Is there a way to define orthogonality of a vector to a subspace/subset in a normed space? 4 / 23 Let us see a characterization of a vector being orthogonal to a subspace in an inner product space. Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Idea of Birkhoff-James Orthogonality Question: Is there a way to define orthogonality of a vector to a subspace/subset in a normed space? If we can define some kind of orthogonality in a normed space, then we might be able to guess some results from geometric intuition and then try to prove it. 4 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Idea of Birkhoff-James Orthogonality Question: Is there a way to define orthogonality of a vector to a subspace/subset in a normed space? If we can define some kind of orthogonality in a normed space, then we might be able to guess some results from geometric intuition and then try to prove it. Let us see a characterization of a vector being orthogonal to a subspace in an inner product space. 4 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Idea of Birkhoff-James Orthogonality Question: Is there a way to define orthogonality of a vector to a subspace/subset in a normed space? If we can define some kind of orthogonality in a normed space, then we might be able to guess some results from geometric intuition and then try to prove it. Let us see a characterization of a vector being orthogonal to a subspace in an inner product space. 4 / 23 James, in 1945, introduced a notion of orthogonality in any normed space using this characterization. Definition (Birkhoff-James Orthogonality) Let (V ; k·k) be a normed space. Then a vector v is said to be Birkhoff-James orthogonal to a subspace W of V if kvk ≤ kv − wk for all w 2 W . Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality Theorem: In an inner product space V , v is orthogonal to a subspace W of V if and only if kvk ≤ kv − wk for all w 2 W . 5 / 23 Definition (Birkhoff-James Orthogonality) Let (V ; k·k) be a normed space. Then a vector v is said to be Birkhoff-James orthogonal to a subspace W of V if kvk ≤ kv − wk for all w 2 W . Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality Theorem: In an inner product space V , v is orthogonal to a subspace W of V if and only if kvk ≤ kv − wk for all w 2 W . James, in 1945, introduced a notion of orthogonality in any normed space using this characterization. 5 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality Theorem: In an inner product space V , v is orthogonal to a subspace W of V if and only if kvk ≤ kv − wk for all w 2 W . James, in 1945, introduced a notion of orthogonality in any normed space using this characterization. Definition (Birkhoff-James Orthogonality) Let (V ; k·k) be a normed space. Then a vector v is said to be Birkhoff-James orthogonal to a subspace W of V if kvk ≤ kv − wk for all w 2 W . 5 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality Theorem: In an inner product space V , v is orthogonal to a subspace W of V if and only if kvk ≤ kv − wk for all w 2 W . James, in 1945, introduced a notion of orthogonality in any normed space using this characterization. Definition (Birkhoff-James Orthogonality) Let (V ; k·k) be a normed space. Then a vector v is said to be Birkhoff-James orthogonal to a subspace W of V if kvk ≤ kv − wk for all w 2 W . 5 / 23 Now, kvk ≤ kv − wk for all w 2 W is same as dist(v; W ) = kvk. Thus, a vector v is Birkhoff-James orthogonal to W if and only if dist(v; W ) is attained at 0. This is abbreviated as v has a best approximation to W at 0. Definition An element w0 2 W is said to be best approximation to v in W if and only if dist(v; W ) = kv − w0k : Equivalently, v − w0 is Birkhoff-James orthogonal to W . So results about Birkhoff-James orthogonality give results for best approximations of a point to a subspace and vice versa. Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality and best approximation Let W be a subspace of normed space V . 6 / 23 Thus, a vector v is Birkhoff-James orthogonal to W if and only if dist(v; W ) is attained at 0. This is abbreviated as v has a best approximation to W at 0. Definition An element w0 2 W is said to be best approximation to v in W if and only if dist(v; W ) = kv − w0k : Equivalently, v − w0 is Birkhoff-James orthogonal to W . So results about Birkhoff-James orthogonality give results for best approximations of a point to a subspace and vice versa. Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality and best approximation Let W be a subspace of normed space V . Now, kvk ≤ kv − wk for all w 2 W is same as dist(v; W ) = kvk. 6 / 23 This is abbreviated as v has a best approximation to W at 0. Definition An element w0 2 W is said to be best approximation to v in W if and only if dist(v; W ) = kv − w0k : Equivalently, v − w0 is Birkhoff-James orthogonal to W . So results about Birkhoff-James orthogonality give results for best approximations of a point to a subspace and vice versa. Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality and best approximation Let W be a subspace of normed space V . Now, kvk ≤ kv − wk for all w 2 W is same as dist(v; W ) = kvk. Thus, a vector v is Birkhoff-James orthogonal to W if and only if dist(v; W ) is attained at 0. 6 / 23 Definition An element w0 2 W is said to be best approximation to v in W if and only if dist(v; W ) = kv − w0k : Equivalently, v − w0 is Birkhoff-James orthogonal to W . So results about Birkhoff-James orthogonality give results for best approximations of a point to a subspace and vice versa. Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality and best approximation Let W be a subspace of normed space V . Now, kvk ≤ kv − wk for all w 2 W is same as dist(v; W ) = kvk. Thus, a vector v is Birkhoff-James orthogonal to W if and only if dist(v; W ) is attained at 0. This is abbreviated as v has a best approximation to W at 0. 6 / 23 So results about Birkhoff-James orthogonality give results for best approximations of a point to a subspace and vice versa. Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality and best approximation Let W be a subspace of normed space V . Now, kvk ≤ kv − wk for all w 2 W is same as dist(v; W ) = kvk. Thus, a vector v is Birkhoff-James orthogonal to W if and only if dist(v; W ) is attained at 0. This is abbreviated as v has a best approximation to W at 0. Definition An element w0 2 W is said to be best approximation to v in W if and only if dist(v; W ) = kv − w0k : Equivalently, v − w0 is Birkhoff-James orthogonal to W . 6 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality and best approximation Let W be a subspace of normed space V . Now, kvk ≤ kv − wk for all w 2 W is same as dist(v; W ) = kvk.
Load more

Recommended publications
  • Introduction to Linear Bialgebra
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of New Mexico University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2005 INTRODUCTION TO LINEAR BIALGEBRA Florentin Smarandache University of New Mexico, [email protected] W.B. Vasantha Kandasamy K. Ilanthenral Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Analysis Commons, Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation Smarandache, Florentin; W.B. Vasantha Kandasamy; and K. Ilanthenral. "INTRODUCTION TO LINEAR BIALGEBRA." (2005). https://digitalrepository.unm.edu/math_fsp/232 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. INTRODUCTION TO LINEAR BIALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail: [email protected] K. Ilanthenral Editor, Maths Tiger, Quarterly Journal Flat No.11, Mayura Park, 16, Kazhikundram Main Road, Tharamani, Chennai – 600 113, India e-mail: [email protected] HEXIS Phoenix, Arizona 2005 1 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Introduction to Clifford's Geometric Algebra
    解説：特集 コンピューテーショナル・インテリジェンスの新展開 —クリフォード代数表現など高次元表現を中心として— Introduction to Clifford’s Geometric Algebra Eckhard HITZER＊ ＊Department of Applied Physics, University of Fukui, Fukui, Japan ＊E-mail: [email protected] Key Words：hypercomplex algebra, hypercomplex analysis, geometry, science, engineering. JL 0004/12/5104–0338 C 2012 SICE erty 15), 21) that any isometry4 from the vector space V Abstract into an inner-product algebra5 A over the field6 K can Geometric algebra was initiated by W.K. Clifford over be uniquely extended to an isometry7 from the Clifford 130 years ago. It unifies all branches of physics, and algebra Cl(V )intoA. The Clifford algebra Cl(V )is has found rich applications in robotics, signal process- the unique associative and multilinear algebra with this ing, ray tracing, virtual reality, computer vision, vec- property. Thus if we wish to generalize methods from tor field processing, tracking, geographic information algebra, analysis, calculus, differential geometry (etc.) systems and neural computing. This tutorial explains of real numbers, complex numbers, and quaternion al- the basics of geometric algebra, with concrete exam- gebra to vector spaces and multivector spaces (which ples of the plane, of 3D space, of spacetime, and the include additional elements representing 2D up to nD popular conformal model. Geometric algebras are ideal subspaces, i.e. plane elements up to hypervolume ele- to represent geometric transformations in the general ments), the study of Clifford algebras becomes unavoid- framework of Clifford groups (also called versor or Lip- able. Indeed repeatedly and independently a long list of schitz groups). Geometric (algebra based) calculus al- Clifford algebras, their subalgebras and in Clifford al- lows e.g., to optimize learning algorithms of Clifford gebras embedded algebras (like octonions 17))ofmany neurons, etc.
    [Show full text]
  • A Bit About Hilbert Spaces
    A Bit About Hilbert Spaces David Rosenberg New York University October 29, 2016 David Rosenberg (New York University ) DS-GA 1003 October 29, 2016 1 / 9 Inner Product Space (or “Pre-Hilbert” Spaces) An inner product space (over reals) is a vector space V and an inner product, which is a mapping h·,·i : V × V ! R that has the following properties 8x,y,z 2 V and a,b 2 R: Symmetry: hx,yi = hy,xi Linearity: hax + by,zi = ahx,zi + b hy,zi Postive-definiteness: hx,xi > 0 and hx,xi = 0 () x = 0. David Rosenberg (New York University ) DS-GA 1003 October 29, 2016 2 / 9 Norm from Inner Product For an inner product space, we define a norm as p kxk = hx,xi. Example Rd with standard Euclidean inner product is an inner product space: hx,yi := xT y 8x,y 2 Rd . Norm is p kxk = xT y. David Rosenberg (New York University ) DS-GA 1003 October 29, 2016 3 / 9 What norms can we get from an inner product? Theorem (Parallelogram Law) A norm kvk can be generated by an inner product on V iff 8x,y 2 V 2kxk2 + 2kyk2 = kx + yk2 + kx - yk2, and if it can, the inner product is given by the polarization identity jjxjj2 + jjyjj2 - jjx - yjj2 hx,yi = . 2 Example d `1 norm on R is NOT generated by an inner product. [Exercise] d Is `2 norm on R generated by an inner product? David Rosenberg (New York University ) DS-GA 1003 October 29, 2016 4 / 9 Pythagorean Theroem Definition Two vectors are orthogonal if hx,yi = 0.
    [Show full text]
  • True False Questions from 4,5,6
    (c)2015 UM Math Dept licensed under a Creative Commons By-NC-SA 4.0 International License. Math 217: True False Practice Professor Karen Smith 1. For every 2 × 2 matrix A, there exists a 2 × 2 matrix B such that det(A + B) 6= det A + det B. Solution note: False! Take A to be the zero matrix for a counterexample. 2. If the change of basis matrix SA!B = ~e4 ~e3 ~e2 ~e1 , then the elements of A are the same as the element of B, but in a different order. Solution note: True. The matrix tells us that the first element of A is the fourth element of B, the second element of basis A is the third element of B, the third element of basis A is the second element of B, and the the fourth element of basis A is the first element of B. 6 6 3. Every linear transformation R ! R is an isomorphism. Solution note: False. The zero map is a a counter example. 4. If matrix B is obtained from A by swapping two rows and det A < det B, then A is invertible. Solution note: True: we only need to know det A 6= 0. But if det A = 0, then det B = − det A = 0, so det A < det B does not hold. 5. If two n × n matrices have the same determinant, they are similar. Solution note: False: The only matrix similar to the zero matrix is itself. But many 1 0 matrices besides the zero matrix have determinant zero, such as : 0 0 6.
    [Show full text]
  • 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.
    [Show full text]
  • 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` .
    [Show full text]
  • The Determinant Inner Product and the Heisenberg Product of Sym(2)
    INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY VOLUME 14 NO. 1 PAGE 1–12 (2021) DOI: HTTPS://DOI.ORG/10.32323/IEJGEO.602178 The determinant inner product and the Heisenberg product of Sym(2) Mircea Crasmareanu (Dedicated to the memory of Prof. Dr. Aurel BEJANCU (1946 - 2020)Cihan Ozgur) ABSTRACT The aim of this work is to introduce and study the nondegenerate inner product < ·; · >det induced by the determinant map on the space Sym(2) of symmetric 2 × 2 real matrices. This symmetric bilinear form of index 2 defines a rational symmetric function on the pairs of rays in the plane and an associated function on the 2-torus can be expressed with the usual Hopf bundle projection 3 2 1 S ! S ( 2 ). Also, the product < ·; · >det is treated with complex numbers by using the Hopf invariant map of Sym(2) and this complex approach yields a Heisenberg product on Sym(2). Moreover, the quadratic equation of critical points for a rational Morse function of height type generates a cosymplectic structure on Sym(2) with the unitary matrix as associated Reeb vector and with the Reeb 1-form being half of the trace map. Keywords: symmetric matrix; determinant; Hopf bundle; Hopf invariant AMS Subject Classification (2020): Primary: 15A15 ; Secondary: 15A24; 30C10; 22E47. 1. Introduction This short paper, whose nature is mainly expository, is dedicated to the memory of Professor Dr. Aurel Bejancu, who was equally a gifted mathematician and a dedicated professor. As we hereby present an in memoriam paper, we feel it would be better if we choose an informal presentation rather than a classical structured discourse, where theorems and propositions are followed by their proofs.
    [Show full text]
  • Inner Product Spaces
    CHAPTER 6 Woman teaching geometry, from a fourteenth-century edition of Euclid’s geometry book. Inner Product Spaces In making the definition of a vector space, we generalized the linear structure (addition and scalar multiplication) of R2 and R3. We ignored other important features, such as the notions of length and angle. These ideas are embedded in the concept we now investigate, inner products. Our standing assumptions are as follows: 6.1 Notation F, V F denotes R or C. V denotes a vector space over F. LEARNING OBJECTIVES FOR THIS CHAPTER Cauchy–Schwarz Inequality Gram–Schmidt Procedure linear functionals on inner product spaces calculating minimum distance to a subspace Linear Algebra Done Right, third edition, by Sheldon Axler 164 CHAPTER 6 Inner Product Spaces 6.A Inner Products and Norms Inner Products To motivate the concept of inner prod- 2 3 x1 , x 2 uct, think of vectors in R and R as x arrows with initial point at the origin. x R2 R3 H L The length of a vector in or is called the norm of x, denoted x . 2 k k Thus for x .x1; x2/ R , we have The length of this vector x is p D2 2 2 x x1 x2 . p 2 2 x1 x2 . k k D C 3 C Similarly, if x .x1; x2; x3/ R , p 2D 2 2 2 then x x1 x2 x3 . k k D C C Even though we cannot draw pictures in higher dimensions, the gener- n n alization to R is obvious: we define the norm of x .x1; : : : ; xn/ R D 2 by p 2 2 x x1 xn : k k D C C The norm is not linear on Rn.
    [Show full text]