Operator Theory and Ill-Posed Problems 9783110960723

Table of contents :
BASIC CONCEPTS
Chapter 1. Set theory
1.1 Sets
1.2 Correspondences
1.3 Relations
1.4 Induction
1.5. Natural numbers
Chapter 2. Algebra
2.1 Abstract algebra
2.2 Linear algebra
2.3 Multilinear algebra
Chapter 3. Calculus
3.1. Limit
3.2. Differential
3.3 Integral
3.4 Analysis on manifolds
OPERATORS
Chapter 4. Linear operators
4.1 Hubert spaces
4.2 Fourier series
4.3. Function spaces
4.4 Fourier transform
4.5 Bounded linear operators
4.6 Compact linear operators
4.7 Self-adjoint operators
4.8 Spectra of operators
4.9. Spectral theorem
4.10. Operator exponential
Chapter 5. Nonlinear operators
5.1 Fixed points
5.2 Saddle points
5.3 Monotonie operators
5.4 Nonlinear contractions
5.5 Degree theory
ILL-POSED PROBLEMS
Chapter 6. Classic problems
6.1 Mathematical description of the laws of physics
6.2 Equations of the first order
6.3 Classification of differential equations of the second order
6.4 Elliptic equations
6.5 Hyperbolic and parabolic equations
6.6 The notion of well-posedness
Chapter 7. Ill-posed problems
7.1 Ill-posed Cauchy problems
7.2 Analytic continuation and interior problems
7.3. Weakly and strongly ill-posed problems. Problems of differentiation
7.4. 7.4 Reducing ill-posed problems to integral equations
Chapter 8. Physical problems leading to ill-posed problems
8.1 Interpretation of measurement data from physical devices
8.2 Interpretation of gravimetric data
8.3 Problems for the diffusion equation
8.4 Determining physical fields from the measurements data
8.5 Tomography
Chapter 9. Operator and integral equations
9.1 Definitions of well-posedness
9.2 Regularization
9.3 Linear operator equations
9.4 Integral equations with weak singularities
9.5 Scalar Volterra equations
9.6Volterra operator equations
Chapter 10. Evolution equations
10.1 Cauchy problem and semigroups of operators
10.2 Equations in a Hilbert space
10.3 Equations with variable operator
10.4 Equations of the second order
10.5 Well-posed and ill-posed Cauchy problems
10.6 Equations with integro-differential operators
Chapter 11. Problems of integral geometry
11.1 Statement of problems of integral geometry
11.2 The Radon problem
11.3 Reconstructing a function from spherical means
11.4 Planar problem of the general form
11.5 Spatial problems of the general form
11.6 Problems of the Volterra type for manifolds invariant with respect to the translation group
11.7 Planar problems of integral geometry with a perturbation
Chapter 12. Inverse problems
12.1Statement of inverse problems
12.2 Inverse dynamic problem. A linearization method
12.3. A general method for studying inverse problems for hyperbolic equations
12.4 The connection between inverse problems for hyperbolic, elliptic, and parabolic equations
12.5 Problems of determining a Riemannian metric
Chapter 13. Several areas of the theory of ill-posed problems, inverse problems, and applications
Bibliography
Index

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Inverse and Ill-Posed Problems Series Operator Theory and Ill-Posed Problems

Also available in the Inverse and Ill-Posed Problems Series: Well-posed, Ill-posed, and Intermediate Problems with Applications Yu.P. Petrov and V.S. Sizikov Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems S.I. Kabanikhin, A.D. Satybaev and M.A. Shishlenin Characterisation of Bio-Particles from Light Scattering V.P. Maltsev and K.A. Semyanov Carleman Estimates for Coefficient Inverse Problems and Numerical Applications /VI.V. Klibanov and A.A Timonov Counterexamples in Optimal Control Theory S.Ya. Serovaiskii Inverse Problems of Mathematical Physics M.M. Lavrentiev, A.V. Avdeev, M.M. Lavrentiev.Jr., and V.l. Priimenko Ill-Posed Boundary-Value Problems S.E. Temirbolat Linear Sobolev Type Equations and Degenerate Semigroups of Operators G.A. Sviridyuk and V.E. Fedorov Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis Editors: M.M. Lavrent'ev and S.I. Kabanikhin Forward and Inverse Problems for Hyperbolic, Elliptic and Mixed Type Equations A.G. Megrabov Nonclassical Linear Volterra Equations of the First Kind A.S. Apartsyn Poorly Visible Media in X-ray Tomography D.S. Anikonov, V.G. Nazarov, and I.V. Prokhorov Dynamical Inverse Problems of Distributed Systems V.l. Maksimov Theory of Linear Ill-Posed Problems and its Applications V.K. Ivanov, V.V. Vasin and V.P. Tanana Ill-Posed Internal Boundary Value Problems for the Biharmonic Equation M.A. Atakhodzhaev Investigation Methods for Inverse Problems V.G. Romanov Operator Theory. Nonclassical Problems S.G. Pyatkov Inverse Problems for Partial Differential Equations Yu.Ya. Belov Method of Spectral Mappings in the Inverse Problem Theory V. Yurko Theory of Linear Optimization I.I. Εremin Integral Geometry and Inverse Problems for Kinetic Equations A.Kh. Amirov Computer Modelling in Tomography and Ill-Posed Problems M.M. Lavrent'ev, S.M. Zerkal and O.E. Trofimov An Introduction to Identification Problems via F unctional Analysis A. Lorenzi Coefficient Inverse Problems for Parabolic Type Equations and Their Application P.G. Danilaev

Inverse Problems for Kinetic and Other Evolution Equations Yu.E. Anikonov Inverse Problems of Wave Processes A.S. Blagoveshchenskii Uniqueness Problems for Degenerating Equations and Nonclassical Problems S P. Shishatskii, A. Asanov and E.R. Atamanov Uniqueness Questions in Reconstruction of Multidimensional Tomography-Type Projection Data V.P. Golubyatnikov Monte Carlo Method for Solving Inverse Problems of Radiation Transfer V.S. Antyufeev Introduction to the Theory of Inverse Problems A.L. ßukhgeim Identification Problems of Wave Phenomena - Theory and Numerics S.I. Kabanikhin and A. Lorenzi Inverse Problems of Electromagnetic Geophysical Fields P.S. Martyshko Composite Type Equations and Inverse Problems A.I. Kozhanov Inverse Problems of Vibrational Spectroscopy A.G. Yagola, I.V. Kochikov, G.M. Kuramshina and Yu.A. Pentin Elements of the Theory of Inverse Problems A.M. Denisov Volterra Equations and Inverse Problems A.L. Bughgeim Small Parameter Method in Multidimensional Inverse Problems A.S. Barashkov Regularization, Uniqueness and Existence of Volterra Equations of the First Kind A. Asanov Methods for Solution of Nonlinear Operator Equations V.P. Tanana Inverse and Ill-Posed Sources Problems Yu.E. Anikonov, B.A. Bubnov and G.N. Erokhin Methods for Solving Operator Equations V.P. Tanana Nonclassical and Inverse Problems for Pseudoparabolic Equations A. Asanov and E.R. Atamanov Formulas in Inverse and Ill-Posed Problems Yu.E. Anikonov Inverse Logarithmic Potential Problem V.G. Cherednichenko Multidimensional Inverse and Ill-Posed Problems for Differential Equations Yu.E. Anikonov Ill-Posed Problems with A Priori Information V.V. Vasin and A.L. Ageev Integral Geometry of Tensor Fields V.A. Sharafutdinov Inverse Problems for Maxwell's Equations V.G. Romanov and S.I. Kabanikhin

INVERSE AND ILL-POSED PROBLEMS SERIES

Operator Theory and Ill-Posed Problems

M.M. Lavrent'ev and L.Ya. Savel'ev

///VSP/// LEIDEN · BOSTON

2006

A C.I.P. record for this book is available from the Library of Congress

ISBN-13: 978-90-6764-448-8 ISBN-10: 90-6764-448-X

© Copyright 2006 by Koninklijke Brill NV, Leiden, The Netherlands. Koninklijke Brill NV incorporates the imprints Brill Academic Publishers, Martinus Nijhoff Publishers and VSP

All rights reserved. No part of this publication may be reproduced, translated, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission from the publisher. Authorization to photocopy items for internal or personal use is granted by Brill provided that the appropriate fees are paid directly to the Copyright Clearance Center, 222 Rosewood Drive, Suite 910, Danvers, MA 01923, USA. Fees are subject to change.

Printed and bound in The

Netherlands.

ν The book is based on the course of lectures on calculus and functional analysis and several special courses given by the authors at Novosibirsk State University. It also includes results of research carried out at the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences. A brief introduction to the language of set theory and elements of abstract, linear, and multilinear algebra is provided. The language of topology is introduced and fundamental concepts of analysis for vector spaces and manifolds are described in detail. The most often used spaces of smooth and generalized functions, their transformations, and the classes of linear and nonlinear operators are considered. Special attention is given to spectral theory and the fixed point theorems. A brief presentation of degree theory is provided. The part devoted to ill-posed problems includes a description of partial differential equations, integral and operator equations, and problems of integral geometry. The book can serve as a textbook or reference on functional analysis. It contains many examples. It can also be of interest to specialists in the above fields.

vi

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

Contents

BASIC CONCEPTS

Chapter 1. Set theory 1.1. Sets

L

2 2

1.1.1. Elements and subsets

2

1.1.2. The algebra of sets

3

1.1.3. Cartesian product

5

1.2. Correspondences

7

1.2.1. Images and inverse images

7

1.2.2. Functions

8

1.2.3. Collections of sets 1.3. Relations

11 14

1.3.1. Reflexivity, transitivity, and symmetry

14

1.3.2. Equivalence

15

1.3.3. Order

17

1.4. Induction

23

1.4.1. Well-ordered sets

23

1.4.2. Discrete sets

24

1.4.3. Zorn's lemma

26

1.5. Natural numbers

27

1.5.1. Decimal natural numbers

27

1.5.2. The isomorphism theorem

28

1.5.3. Countable sets

29

viii

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

Chapter 2. Algebra 2.1. Abstract algebra 2.1.1. Semigroups 2.1.2. Groups 2.1.3. 2.1.4. 2.1.5. 2.2. Linear 2.2.1. 2.2.2.

Rings and Lattices Numbers algebra Vector spaces Linear operators

30 30 30 37 fields

2.2.3. Linear functionals 2.2.4. Scalar products 2.2.5. Normed spaces 2.2.6. Euclidean spaces 2.3. Multilinear algebra 2.3.1. Tensor product 2.3.2. Exterior product Chapter 3. Calculus

55 64 69 75 76 89 102 114 121 130 141 141 146 150

3.1. Limit 3.1.1. Topological spaces 3.1.2. Directed sets 3.1.3. Convergence 3.2. Differential 3.2.1. The definition of the differential 3.2.2. Differentiation rules

150 150 170 177 197 197 206

3.2.3. Lagrange's theorem 3.2.4. Termwise differentiation 3.2.5. Total differentials 3.2.6. Solution of functional equations 3.2.7. Taylor's formula 3.2.8. Local minima 3.2.9. Smooth curves 3.2.10. A simplest variational problem

210 215 217 226 241 250 258 261

Contents 3.3. Integral 3.3.1. Measures 3.3.2. Classical definition of the integral 3.3.3. Limit theorems 3.3.4. Measurable functions 3.3.5. The Fubini and Tonelli theorems 3.3.6. Indefinite integrals 3.4. Analysis on manifolds 3.4.1. Manifolds 3.4.2. The rank theorem 3.4.3. Sard's theorem 3.4.4. Differential forms 3.4.5. The Poincare theorem 3.4.6. Change of variables 3.4.7. Integral over a manifold 3.4.8. The Stokes formula 3.4.9. Map degree 3.4.10. Applications OPERATORS

Chapter 4. Linear operators 4.1. Hilbert spaces 4.1.1. Orthogonal projection 4.1.2. Continuous linear functionals 4.1.3. The spaces C2 = C2(U, μ) 4.2. Fourier series 4.2.1. Fourier coefficients 4.2.2. Isomorphism of Hilbert spaces 4.3. Function spaces 4.3.1. Metric spaces 4.3.2. Smooth functions 4.3.3. Lebesgue spaces 4.3.4. Distributions 4.3.5. Sobolev spaces

ix 263 264 272 289 295 298 316 321 321 326 327 328 331 334 335 342 348 351 353

354 354 354 356 359 363 363 368 369 369 371 376 380 388

χ

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed ... 4.4. Fourier transform 4.4.1. Transforms of rapidly decreasing functions 4.4.2. Transforms of slowly increasing distributions 4.4.3. The Fourier-Plancherel transform 4.4.4. The Fourier-Stieltjes transform 4.4.5. The Radon transform 4.5. Bounded linear operators 4.5.1. Extensions of functionals 4.5.2. Uniform boundedness of operators 4.5.3. Inversion of operators 4.5.4. Closedness of the graph of an operator 4.5.5. Weak compactness 4.6. Compact linear operators 4.6.1. Examples of compact operators 4.6.2. Properties of compact operators 4.6.3. Adjoint operators 4.6.4. Fredholm operators 4.6.5. Fredholm theorems 4.7. Self-adjoint operators 4.7.1. Banach adjoint operators 4.7.2. Hilbert adjoint operators 4.7.3. Hermitian and normal operators 4.7.4. Unitary operators 4.7.5. Positive operators 4.8. Spectra of operators 4.8.1. Classification of spectra 4.8.2. The spectrum of a closed operator 4.8.3. The spectrum of a bounded operator 4.8.4. The spectrum of a compact operator 4.8.5. The spectrum of a self-adjoint operator 4.9. Spectral theorem 4.9.1. Projection measures 4.9.2. Integrals of bounded functions 4.9.3. Integrals of unbounded functions 4.9.4. Spectral theorem 4.9.5. Operator functions

391 391 393 395 395 396 397 397 399 401 403 405 408 408 410 411 414 417 419 419 420 422 423 424 426 426 431 433 435 435 444 445 451 459 462 466

Contents 4.10. Operator exponential

xi 468

4.10.1. Problem formulation

468

4.10.2. Semigroups of operators

470

4.10.3. The Laplace transform

475

4.10.4. Stone's theorem

476

4.10.5. Evolution equations

478

Chapter 5. Nonlinear operators

483

5.1. Fixed points

483

5.1.1. The Brouwer theorem

483

5.1.2. The Tikhonov theorem and the Schauder theorem . . . 487 5.2. Saddle points

490

5.2.1. Kakutani's theorem

490

5.2.2. von Neumann theorem

493

5.3. Monotonie operators

497

5.3.1. Definition and properties

497

5.3.2. Equations with monotonic operators

499

5.4. Nonlinear contractions

501

5.4.1. Contracting semigroups of operators

501

5.4.2. Approximation

503

5.5. Degree theory

504

5.5.1. Finite-dimensional spaces

504

5.5.2. The Leray-Schauder degree

508

ILL-POSED PROBLEMS

Chapter 6. Classic problems

511

512

6.1. Mathematical description of the laws of physics

512

6.2. Equations of the first order

518

6.3. Classification of differential equations of the second order . . . 519 6.4. Elliptic equations

521

6.5. Hyperbolic and parabolic equations

527

6.6. The notion of well-posedness

529

xii

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

Chapter 7. Ill-posed problems 531 7.1. Ill-posed Cauchy problems 531 7.2. Analytic continuation and interior problems 534 7.3. Weakly and strongly ill-posed problems. Problems of differentiation 536 7.4. Reducing ill-posed problems to integral equations 537 Chapter 8. Physical problems leading to ill-posed problems 541 8.1. Interpretation of measurement data from physical devices . . . 541 8.2. Interpretation of gravimetric data 543 8.3. Problems for the diffusion equation 546 8.4. Determining physical fields from the measurements data . . . 547 8.5. Tomography 548 Chapter 9. Operator and integral equations 9.1. Definitions of well-posedness 9.2. Regularization 9.3. Linear operator equations 9.4. Integral equations with weak singularities 9.5. Scalar Volterra equations 9.6. Volterra operator equations

552 552 555 559 564 565 568

Chapter 10. Evolution equations 10.1. Cauchy problem and semigroups of operators 10.2. Equations in a Hilbert space 10.3. Equations with variable operator 10.4. Equations of the second order 10.5. Well-posed and ill-posed Cauchy problems 10.6. Equations with integro-differential operators

571 571 573 577 578 580 581

Chapter 11. Problems of integral geometry 11.1. Statement of problems of integral geometry 11.2. The Radon problem 11.3. Reconstructing a function from spherical means 11.4. Planar problem of the general form 11.5. Spatial problems of the general form

584 584 584 588 594 602

Contents

xiii

11.6. Problems of the Volterra type for manifolds invariant with respect to the translation group 614 11.7. Planar problems of integral geometry with a perturbation . . 618 Chapter 12. Inverse problems 626 12.1. Statement of inverse problems 626 12.2. Inverse dynamic problem. A linearization method 628 12.3. A general method for studying inverse problems for hyperbolic equations 637 12.4. The connection between inverse problems for hyperbolic, elliptic, and parabolic equations 644 12.5. Problems of determining a Riemannian metric 651 Chapter 13. Several areas of the theory of ill-posed problems, inverse problems, and applications

659

Bibliography

662

Index

673

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed .

Preface The book consists of three major parts. The first two parts, written by L.Ya. Savel'ev, deal with general mathematical concepts and certain areas of operator theory. The third part, written by Μ. M. Lavrent'ev, is devoted to ill-posed problems. It can be read independently of the first two parts and presents a good example of applying the methods of calculus and functional analysis. The book is based on the lectures given by the authors at Novosibirsk State University. The part "Basic Concepts" briefly introduces the language of set theory and concepts of abstract, linear and multilinear algebra. We also introduce the language of topology and consider fundamental concepts of calculus: the limit, the differential, and the integral. A special section is devoted to analysis on manifolds. The part "Operators" describes the most important function spaces and operator classes for both linear and nonlinear operators. Different kinds of generalized functions and their transformations are considered. Elements of the theory of linear operators are presented. Spectral theory is given a special focus. We prove main theorems on stationary points of nonlinear transformations and briefly introduce the theory of mapping degree. The part "Ill-Posed Problems" is devoted to problems of mathematical physics, integral and operator equations, evolution equations and problems of integral geometry. It also deals with problems of analytic continuation. Detailed coverage of the subjects and numerous examples and exercises make it possible to use the book as a textbook on some areas of calculus and functional analysis. It can also be used as a reference textbook because of the extensive scope and detailed references with comments. Several sections contain new research results, namely, a description of a general model of linear continuous extension of a vector measure to the integral. A uniqueness theorem is proved for a new type of equations with

xvi

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed ...

integro-differential operators. Examples in which the Radon problem is illposed are given. The problem is proved to be weakly ill-posed. The problem with incomplete initial data is analyzed. A uniqueness theorem for a special problem of integral geometry is proved. The above results were discussed at the seminars in the Sobolev Institute of Mathematics (Novosibirsk). The authors wish to thank their colleagues for valuable criticism and advice.

BASIC CONCEPTS

This part of the book describes the areas of set theory, abstract and linear algebra, and geometry of spaces that are widely used in calculus and functional analysis. The chapter "Set Theory" presents the abstract mathematical language used in modern mathematical analysis. The necessary notions axe introduced and useful facts from set theory are provided. The abstract mathematical language makes it possible to formulate propositions in a simple form and make arguments concise and clear. The abstract character of the language is compensated by providing specific examples. The logical connectives if... then, only i f , if and only if (or equivalent) are often denoted by the arrows -ΦΦ-, respectively. The symbols V and 3 often stand for the expressions for all and for some. The general principle of induction and the equivalent propositions used in the proofs of many important theorems are formulated. The chapter "Algebra" includes elements of abstract, linear, and multilinear algebra. It focuses on the areas of these theories that are often used in calculus and functional analysis. Considerable attention is given to algebraic structures, namely semigroups, groups, rings, fields, vector spaces, and algebras. Normed and Euclidean spaces and their geometry are described. A separate section is devoted to elements of multilinear algebra and tensor calculus. The chapter "Calculus" is devoted to limit theory, differential and integral calculus. Directional convergence in topological spaces, differentiation in vector spaces, and integration with respect to a measure are described. A separate section is devoted to analysis on manifolds.

Chapter 1. Set theory Throughout the book we will use naive set theory, which assumes that sets with all the necessary properties exist. The chapter describes the formal language of set theory and operations on sets. 1.1.

SETS

There is no formal definition of a set. It is assumed that properties used to construct sets are not contradictory and the sets with these properties are well defined. 1.1.1.

Elements and subsets

Each nonempty set is defined by its elements. It is also assumed that there exists an empty set, which is a set with no elements. Sets are usually denoted by capital letters, while their elements are denoted by small letters. 1. We indicate that an element α belongs to a set Ε by writing α € Ε. The negation of this statement is written as ο ^ E. A. set Ε defined by enumerating all of its elements a, b, c, ... is said to consist of elements a, b, c, ..., which is denoted by Ε = {a,b,c,... }. The order in which the elements axe put together does not matter. Two sets are considered to be equal if they consist of the same elements. If sets A and Β are equal, we write A = B. The negation of this statement is written as Α φ Β . Sets consisting of exactly one element are called singletons and are often identified with their elements.

Chapter 1. Set theory

3

2. If every element of a set A belongs to a set B, then A is called a subset of B. Alternatively, we say that Β includes A (B D A), or A is included in Β {A C B). Otherwise, if Β contains no elements of A, we write Β A or A Β. If AC Β and Αφ B, then the inclusion of A in Β is called proper and designated as A C Β or Β D A. Every set Ε includes the empty set, which is denoted by 0 . From the definitions, it follows that A = Β if and only if A C Β and Β CA. For a set E, it is convenient to designate its subsets by {χ € Ε | . . . } , where the vertical bar precedes the description of a certain property defining a given subset. Some notations use colon instead of the vertical bar. If it is obvious from the context that χ is an element of a set E, then we write χ instead of χ € Ε. Nonempty subsets of a set Ε that are not equal to Ε are called proper subsets of E. For any set E, there exists a class V = V(E) of all subsets of E. Here the term class means sets whose elements are sets. It is introduced for the sake of convenience in order to avoid phrases such as "a set of sets" or "a set of subsets". Example. Consider the set of digits F = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } . The set of even digits A = { 0 , 2 , 4 , 6 , 8 } and the set of odd digits Β = { 1 , 3 , 5 , 7 , 9 } are subsets of F. The set A — { 2 , 4 , 6 , 8 } is a proper subset of A. Neither A nor Β includes the set D = {0,1}. 1.1.2.

The algebra of sets

The operations of union, intersection, and set difference are defined on the class V = V{E) of all subsets of a set E. 1. Let X, Y, and Ζ be sets of class V. The set X U Υ = {ζ | χ G X or y e Υ} € V, consisting of elements that belong to X or Y is called the union of X and Y. The set Χ Π Υ = {χ \ χ € X and y € Y} e V, consisting of elements that belong to both X and Y is called the intersection of X and Y. The set Υ \ X = {y \ y € Y and y £ X} e V, consisting of elements that belong to Y and do not belong to X is called the complement of X with respect to Y. The complement of X with respect to Ε is called simply the complement X and denoted by X' or Xc. Obviously, Y\X = Y \ ( X n F ) = ΥΓ\Χ'.

4

Μ. Lavrent'ev

and L. Savel'ev.

Operator

theory

and ill-posed

...

2. If Χ Π Y = 0 , then the sets X and Y said to be non-intersecting or disjoint. It follows from the definitions that Χ Π Y = 0 is equivalent to γ

c r .

A class is called disjoint or pairwise disjoint if any two distinct sets in it are disjoint. From the definition, it follows that the empty class and the class containing exactly one set are disjoint. 3. The operations of union, intersection and set difference satisfy the following easily verifiable conditions:

—

—

XUX'

=

E,

χ υχ

=

χ,

(1)

= i

(2)

= Ynx

(3)

idempotency, = Y U X ,

xnY

commutativity;

(χ υ γ) υ ζ = χ υ (γ υ ζ ) , —

= 0

partition·,

XUY —

ΧΠΧ'

(χ Γ) Υ) π ζ = χ η (υ π ζ)

associativity, (χ υ Υ) η ζ — (χ η ζ) υ (Υ π ζ ) , (χ η γ) υ ζ = (χ υ ζ) η (χ υ ζ)

—

x,

x n ( x u Y) = x

(6)

γ',

(.χ η γ)'

(7)

absorption; (χ υ γ)'

—

= x ' n

= χ' υ γ'

duality, ( x y = χ

—

(5)

distributivity, x u ( x n Y ) =

—

(4)

involution.

The class V with operations U, Π, ' is a boolean algebra of sets.

(8)

5

Chapter 1. Set theory

4. The operations of union and intersection can be generalized. Consider an arbitrary class C C P . The set \jC = {x\xedoT

some C e C} € V

of all elements that belong to at least one set of class C is called the union of sets of the class C. For a set A, every class C such that UC includes A is called a cover of A. A disjoint class V such that UV = A is called a partition of A (the usual assumption being that all sets in V are nonempty). The set Π C = {x I χ e C for every C

eC}eV,

is called the intersection of sets of the class C. For example, for a class C consisting of sets A,B,C,..., intersection are written as AußuCu...,

their union and

ANBNCF]....

In particular, if C — {A, B}, then

U{A,B}

= AUB,

D{A,B}

=

ADB,

which follows from the definition of union and intersection, with X = A, Y = B. 5. Example. For the sets A, Β, C, D, F in the example of Section 1.1.1, the following statements are true:

AU Β = F, AC\B = 0, A! = Β, A\C = {0}, A\D = C, ANC = C, BUC\JD

= F,

BncnD

B'= A, A n D = {0}, = 0.

Remark. We do not justify the well-posedness of the above definitions since we use naive set theory. In axiomatic theories, it is ensured by the axioms (see Kuratowski and Mostowski, 1970, Chapter. II).

1.1.3.

Cartesian product

The notion of the Cartesian product is associated with geometry rather than algebra. It formalizes the transition from the line to the plane and the

6

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

multidimensional space. The definition of the Cartesian product is based on the definition of an ordered pair. 1. Consider arbitrary elements α € A and b ζ B, singletons { a } and {6}, and the set {a, 6}. The class (a, b) = { { a } , {a, 6}} with elements { a } and {a, 6} is called the ordered pair whose ßrst element is α and second is b. . We emphasize that this definition uses only the notion of a set. The element ο Ε Α, α φ b, is labeled as the ßrst because { a } € { { a } , {a, b}}, but {b} { { a } , {a, 6}}. For brevity, we will call an ordered pair simply a pair and write ab instead of (a, b). Let χ Ε A, y Ε B. Then the ordered pairs (x, y) and (a, b) are equal if and only if their first elements χ and a are equal and the same is true for their second elements y and b. Exercise. Prove the foregoing statement. If α φ b, then (a, 6) φ (b,a). By definition, (a, a) = { { a } } for every α Ε ΑΠ B. 2. The set Α χ Β = {{x,y) | x Ε A, y Ε B} consisting of all ordered pairs whose first elements belong to A and whose second elements belong to Β is called the Cartesian product of the sets A and Β. Ii Αφ Β, where Α φ 0 and Β φ 0, then Αχ Β φ Β χ Α. Example. Let A = Β = F = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } . Then the Cartesian product Α χ Β = F χ F consists of the ordered pairs 0 0 , 0 1 , . . . ,09, 10,11, . . . , 1 9 , . . . , 9 0 , 9 1 , . . . , 9 9 . 3. Nonempty classes whose elements are nonempty will be called nondegenerate. Consider a nondegenerate disjoint class V. A sample of V is a set Y consisting of elements of sets in V such that for every set Χ Ε T> there is exactly one element y G Y that belongs to X. We assume that the Axiom of choice holds. Axiom of Choice. There exists a sample of any nondegenerate class of sets.

disjoint

The set f j V of all samples of V is called the Cartesian product of the sets of the class T>. The axiom of choice states that Π Τ> φ 0 for every nondegenerate disjoint class T>. For a class T> whose elements are sets A, B,C,..., the Cartesian product Π V is also written as Α χ Β χ C x · · •.

Chapter 1. Set theory

7

Example. Let F = {0,1,2,3,4,5,6,7,8,9} and let V be the class consisting of the sets A = { 1 } x F, Β = { 2 } χ F, C = { 3 } χ F. Then J[V

= { { ( 1 , x), (2, y), (3, ζ)} \ X, y, Z € F}.

Elements of F x F x F can be conveniently written in the form of a sequence xyz = 000,001,..., 999.

1.2.

CORRESPONDENCES

Correspondences are formally defined as sets of ordered pairs. The meaning of this notion is intuitively clear. 1.2.1.

Images and inverse images

For any sets A and Β, every subset S of the Cartesian product Α χ Β is called a correspondence between A and B. If (x, y) G S, this is often written in the form xSy or S : χ —> y. Similarly, S C Α χ Β is often written as S :A^B. 1. For χ G A, the set S(x) — {y | (x, y) G 5 } C Β is called the image of the element χ under the correspondence S. This image may be empty. For X C A, the union S{X) = U { S ( x ) \ χ G X } of images S(x) of all elements χ 6 X is called the image of the set X under the correspondence S. In particular, the set 5 ( A ) C Β is called the image or the range of the correspondence S, also denoted by Ran S. 2. The correspondence = {(y, χ) | (x, y) G S } C Β χ A is called 1 the inverse of S. The image S~ (y) C A of an element y € Β under is called the inverse image of y under S. This inverse image may be empty. The image S~1(Y) C A of a set Y C Β under S " 1 is called the inverse image of Y under S. It is obvious that 5 _ 1 ( y ) = U{5 _1 (?/) | y 6 Y}. The set S~l(B) C A is called the domain of deßnition of S. The correspondence S is said to be defined on S~1(B), this set being denoted by Dom S and called simply the domain of S. 3. It follows from the definitions that Dom 5 = {χ € A I y € S{x) for some y G B}, Ran S = {y € Β \y G S(x) for some χ G A}. The condition in the first equality means that S(x) φ 0, and the condition in the second one means that S~1(y) φ 0 . ) If Dom S = A, then the

8

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

correspondence S is said to be defined everywhere. If Ran S = B, then the correspondence S is called a covering correspondence. Let S C Τ C A x B. Then S is called a restriction of the correspondence T, and Τ is called an extension of the correspondence S. The inclusion S C Γ is equivalent to the inclusions S C Dom Τ and S(x) C T(x) for every χ € Dom S. 4. Consider sets A, Β, C and correspondences SCAxB,TCBxC. The correspondence TS C Ax C defined by the formula (TS)(x) = T(S(x)) (x € A) is called the composition of correspondences S and T. The domain Dom (TS) is empty if Ran S and Dom Τ do not intersect. It follows from the definitions that (TS)~ l = S ^ T " 1 . The composition of the correspondences S and Τ can be represented in a diagram. Diagrams are especially convenient to represent compositions of more than two correspondences. The term product is often used as a synonym to composition.

Example. Let F = {0,1,2,3,4,5,6,7,8,9} and FxF = {00,01,...,09; 10,11,..., 19; . . . 90,91,..., 99}. The correspondence S = {01,02,..., 09, 12,..., 19, . . . , 89} C JF χ F is a strict order on F. In this case, y G S(x) means that y > x. The correspondence S~l = {10,21,20,..., 9 8 , . . . , 91,90} C F x F is the inverse of the strict order on F. Then y € S~1(x) means that y < x. The correspondence Τ = {00,11,24,39} C F x F associates every number with its square: ζ 6 T(y) means ζ = y2. We have T(S(0)) = Γ({1,2,3, . . . , 9 } ) = {1,4,9}, Γ(5(1)) = Γ({2,3,...,9}) = {4,9}, T(S(2)) = T ( { 3 , . . . , 9}) = {9}, T(S(S)) = · · · = T({9}) = 0 . Therefore, TS = {01,04,09, 14, 19,29} CFxF.

1.2.2.

Functions

Correspondences for which the image of every element is either empty or consists of exactly one element are called functions.

Chapter 1. Set theory

9

1. A correspondence / C Α χ Β such that every χ € Dom / C A has a unique image y 6 Ran / C Β is said to be a many-to-one correspondence. A many-to-one correspondence / C Α χ Β is also called a function from A to Β or a function defined on A with values in B. It is conventional to write y = f(x) instead of {y} = fix)· The element y is called the vaiue of / corresponding to x. Exercise. Prove that the composition of functions is a function. Remark. By definition, a function is a set of ordered pairs. From the definition, it is intuitively clear that a function is equivalent to its graph. Sometimes a graph / C Α χ Β is defined independently and a correspondence from A to Β is defined as a triple ( A , B , f ) . This provides a way to treat functions as sets, which often appears to be convenient. 2. Functions defined everywhere are called maps. A function / C Α χ Β with Dom / = A is said to map A to Β. In addition, if Ran / = B, then / is said to map A onto B. If / maps A to B, it is conventional to write / : f A —> Β or A A Β instead of / C Α χ Β. The set of all maps of A to Β is denoted by BA. 3. The correspondence / - 1 C Β χ A inverse to / C Α χ Β is not necessarily a function. If / is a many-to-one correspondence and so is its inverse, then / is called a one-to-one correspondence. A one-to-one map from A to Β is called an injection or embedding. A one-to-one map from A onto Β is called a surjection. It is also called an isomorphism from A onto B. If there exists an isomorphism between two sets, these sets are said to be isomorphic. An isomorphism from A onto Β is also called a substitution of A into B, and an isomorphism from A onto itself is called an automorphism or a permutation of A. All isomorphisms are one-to-one surjections. Examples. 1) Let F = {0,1,2,3,4,5,6,7,8,9}. Then F χ F = {00, 0 1 , . . . , 09; 10, 11,..., 19; . . . 90,91,..., 99}. The correspondence / = {00,12,24, 36,48} is a function with domain Dom / = {0,1,2,3,4} and range Ran / = {0,2,4,6,8}. The inverse correspondence f ~ l = {00,21, 42, 63,84} is also a function with domain D o m / - 1 = R a n / and range R a n / - 1 = D o m / . It is clear that / is a one-to-one correspondence. It maps the set A = {0,1,2,3,4} onto Β = {0,2,4,6,8}. These sets are isomorphic. 2) The map X —• y (X G V, y Ε X) of a disjoint class T> onto its sample Y is an isomorphism (see Section 1.1.3).

10

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed ...

Remark. We emphasize the importance of specifying the correspondences together with the sets involved. The set of pairs / in the preceding example is included in both Cartesian products Α χ Β and F x F. At the same time, the correspondence / in Α χ Β is an isomorphism, whereas it is not so in F χ F. 4. For sets A, B, C, consider embeddings / : A —• B, g : Β —> C, and their composition h = gf : A C. Proposition. The composition h of embeddings f and g is an embedding. Proof. We now prove that h is an embedding. Let h(x) = h(y) for some x, y Ε A. This means that g{f{x)) = g(f(y))· We have f(x) = f(y) because g is a one-to-one correspondence. Since / is a one-to-one correspondence, it follows that χ = y. The composition h is a one-to-one correspondence. • Corollary. The composition h of isomorphisms f and g is an isomorphism. Proof. Based on the above proposition, it suffices to show that the composition h = gf of the surjections g and / is a surjection. Take any ζ Ε C. Since g is a surjection, there exists an element y £ Β such that g(y) = z. Since / is a surjection, there exists an element χ Ε A such that f(x) — y. Consequently, h(x) = g(f(x)) = ζ and thus h is a surjection. • 5. It is often convenient to represent sets by functions called indicators. Consider a set U, the class V = V(U) of all subsets of U, and the set Β = {0,1}. For every Χ Ε V, the function indX : U -> Β is defined, with values indX(u) = 1 for u Ε X and indX(u) = 0 for u £ X. Functions defined on U with values in Β are called indicators on U. The function indX is called the indicator of the subset X ofU. The set of all indicators on U is denoted by X = I{U). The map ind from V to X, which associates every set Χ Ε V with its indicator ind Χ Ε Τ, is an isomorphism from V onto I. The isomorphism ind :V—>Z associates sets in V with their indicators in J . Exercise. Prove that ind : V —> X is an isomorphism.

11

Chapter 1. Set theory

Remark. We have demonstrated that functions can be thought of as sets of a specific kind, while sets can be thought of as functions of a specific kind. 6. Functions can be thought of as operators of a general kind, or abstract operators. All operators considered in the following sections have specific domains and ranges, as well as specific properties of the correspondences between the domain and the range. All general definitions and statements for correspondences and functions are valid in the operator theory. The properties of images, inverse images, and compositions are of special importance in this theory. Complex compositions can be conveniently represented using diagrams with arrows. 1.2.3.

Collections of sets

The term collection is a synonym for the term function. It is used in a specific notation for values of functions that involves indices, which often appears to be convenient. 1. Consider a set I of elements called indices. Any map / : I —> Β that maps I into Β will be called a collection of elements of Β indexed by I. We will use the designation fi (i € I) for f(i), and ( f i ) for /. Thus, / = (ft). Whenever necessary, the index set is specified: ( f i ) (i e /). For every collection of sets X , G V = V(E), and the intersection nXf.

we define the union UXi

UXi = {χ I χ € Xi for some i € I}, Π Xi = {χ \ χ Ε Xi for every i € I}. These definitions agree with those for the union and intersection of a class of sets given in Section 1.1.2. The index set is specified whenever necessary: Uie/> r w From the definitions, it follows that

(J Xi

= 0,

fi Xi

= E

(i

€ 0)

for subsets of the collection Ε indexed by the empty set I . A collection whose index set is empty is called an empty collection. For a set A, a collection of nonempty sets Xi such that A is included in X = IJ Xi is called a cover of A. If X = A and the sets Xi are pairwise disjoint, then the cover (X;) is called a set partition of A.

12

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

Every pairwise disjoint class V of nonempty sets X C Ε can be thought of as a collection — the identity map in V, which maps V onto itself. In this case, the sets X 6 T> serve as indices of themselves. If (J V = E, then V is also called a set partition of E. For the sake of simplicity, the requirement for elements of set partitions to be nonempty is sometimes omitted. The Cartesian product of the disjoint class V consisting of sets {i} χ Xi = {(i, Xi) I Xi € Xi\ (i Ε I) is called the Cartesian product of the collection Xi (i G I) and denoted by f ] X i or \ \ i e i Xi- Its elements are collections χ = (xj) of Xi € X{ {i € I): J J X i = {x = (Xi) I Xi € Xi for every i € /}; or, equivalently, functions / : I —* X such that f(i) € Xi for every index iE I: Y[Xi = { f : I - * X \ /(f) € Xi for every i € I}. From the definition, it follows that the Cartesian product of an empty collection is empty. 2. The main properties of operations defined for collections of sets are easy to verify. They can be represented as identities. If A, B, and C are index sets, Xa (a 6 A) and Υβ (β € Β) are collections of sets, ρ : A —> A is a permutation, A(7) (7 € C) is a collection of subsets of A that is a cover of A, then the following properties hold: U Xp(a) = (J XCL, Π -Χρ(α) = Π Xa α£Α aeA aeA α€Α

(1)

— commutativity-,

υ*»=υ( υ

aeA — associativity;

7€C

α€Α(7)

ru-=n η

aeA

= = —

distributivity;

X

7ec aeA(7)

U (a,ß)eAxB

'

(2)

(Xar\Yß),

Π (Χα U Υβ) (a,ß)eAxB

^

Chapter 1. Set

^aeA

'

αeA

^aeA

'

^ßeB

13

theory

^ αβΑ

'

(4)

aeA

duality;

( π

χα) χ ( π

^•aeA

'

'

(a,ß)eAxB

Υβ) =

Π

\ßeB

'

(5) χ

γ

β)

(a,ß)eAxB

cartesian distributivity. If A = B, then the foregoing identity can be simplified as follows: Π XQ) x ( Π Υα) = Π (Χα Χ Υα)· aeA ' ^ α€Α ' aeA

(6)

Exercise. Prove the above identities. 4. Consider sets E, F, and Y C F, collections Xi C Ε and Yj C F, and a map / : Ε F. The following equalities hold for images and inverse images: f(UXi) f-'iuYj)

= ur^Yj),

rHnYj)

ΓΙ(Υ') For the image of intersection,

= Uf(Xi)·,

(7) = n/- ^·); 1

= Γ\Υ)'·

(8) (9)

only the following inclusion is true:

f(nXi)

c nf(Xi).

(10)

Exercise. Prove equalities (7)-(10). Example. Let Ε = F = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } , Χτ = { 1 , 3 , 5 , 7 , 9 } , X2 = { 0 , 2 , 4 , 6 , 8 } and let / : Ε —> F be defined by the equalities / ( 0 ) = / ( I ) = 1, / ( 2 ) - / ( 3 ) = 3, / ( 4 ) = / ( 5 ) = 5, / ( 6 ) = / ( 7 ) = 7, / ( 8 ) = / ( 9 ) = 9. Then f(X{) = f(X2) = Xu Χι η = 0 and η X2) = / ( 0 ) = 0 C l i = f(X i) = f(X2) = Xi.

14

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

Let Yi = {1,2,3,4,5}, Y2 = {2,3,4,5,6}, Y3 = {3,4,5,6,7}. We have r (Y1) = {0,1,2,3,4,5}, r 1 ^ ) = {2,3,4,5}, / ^ ( Y s ) = {2,3,4,5,6,7} and 1

Yi η y 2 η r 3 = {3,4,5}, Therefore, Γ

1

rHYi) η Γ

1

^ ) η Γ ι ( Ή ) = {2,3,4,5}.

fr Π Υ2 Π Υ3) = {2,3,4,5} = Γ

1

(Ή) Π f~ 1 (Y 2 ) Γ) Γ

1

^).

This example illustrates identities (8) and inclusion (10). 6. The union U and intersection Π of a collection of sets can be viewed as abstract operators. Their range is V and their domain of definition is the set F(I, V) of maps that map an index set I to the class V = V{E) of all subsets of a given set E. For every map / : I —• V, its images under the operators of union and intersection are U/ = U, fi £ V and Π/ = f]t fi & V, respectively. Equalities (1)—(10) represent the properties of the operators U and Π. 1.3.

RELATIONS

Correspondences that are subsets of Ε χ Ε are called relations on E. Important examples of relations are equivalence and order. Note that any map from A to Β is a relation on Ε = AUB. 1.3.1.

Reflexivity, transitivity, and symmetry

The properties of reflexivity, transitivity, and symmetry are very common and can be considered to be the most important. 1. A relation R C Ε χ Ε such that χ Ε R(x) for every χ Ε Ε is said to be reßexive. In a reflexive relation, every element of Ε is related to itself (x x) and, maybe, other elements. A relation R C Ε χ Ε such that y € R(x) and ζ € R(y) implies ζ G R(x) for every x, y, ζ € Ε is said to be transitive. In other words, in a transitive relation, if χ is related to y, then χ is related to all elements that y is related to: A relation R C Ε χ Ε such that y e R(x) implies χ € R{y) for all x,y Ε Ε is said to be symmetric: χ y y —> x. Every symmetric relation coincides with its inverse R = R~1. A relation R C Ε χ Ε such that y Ε R(x) and χ Ε R(y) implies χ = y for all x, y Ε Ε is said to beantisymmetric: x-+y,y-+x=$-x = y. Under an antisymmetric relation, only equal elements are related to each other.

Chapter 1. Set theory

15

2. Examples. The relations S and Τ in the example of Subsection 1.2.1 are transitive and antisymmetric, and so is their composition TS. Consider the relation R = {00,11,22,33,44,55,66,77,88,99}, which is the equality relation for digits. It is reflexive, transitive, and symmetric. The order relation R U S for digits is reflexive and transitive. The relation / in Example 1 of Subsection 1.2.2 is not transitive. Indeed, if y = 2x and 2 = 2y, then ζ φ 2x for χ φ 0. 1.3.2.

Equivalence

A reflexive, transitive, and symmetric relation is called an equivalence relation. If R is an equivalence relation on E, then the notation χ ~ y is often used instead of y G R(x) and χ —> y. 1. The following theorem provides a criterion for equivalence relations. Theorem. Every equivalence relation on a given set defines a partition of this set. Conversely, every partition of a given set introduces an equivalence relation on it. Proof. Let R be an equivalence relation on Ε and T> is the class of all images R(x) of elements χ G E. It is required to prove that V is a partition of E. Since R is reflexive, χ G R(x) and therefore V is a cover of E. Let a,b G Ε, χ G R(a), y G R(b), and suppose that there exists an element c € R(a) Π R(b), i.e., χ ~ a, y ~ b and c ~ a, c ~ b. Since R is symmetric and transitive, we have c~ a, c~b=>a~c, c~b=>a~b and χ ~ a, a^b^x^b&xE R(b) R(a) C R(b). Similarly, y~b, a~b=>y~b, b~a=>y~a such that every element χ € Ε is related to the set X &T> that covers x. By definition, R is the union of cartesian squares X x X o f all sets X € V. Since χ G X = R{x), it follows that R is reflexive. If y 6 X = R(x), then X = R(y), χ € R(y) and therefore R is symmetric. Since R(x) = R(y) for any y € R(x), we have 2 G R(x) for any 2 G R{y), which means that R is transitive. Thus, R is an equivalence relation. • The partition of Ε introduced by the equivalence relation R is called a factor class obtained by factoring Ε with respect to R and is denoted by

16

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

E/R. The elements of E/R are called factor sets. Any two elements that belong to the same factor set are equivalent. 2. Examples. 1) The simplest example of an equivalence relation is equality on a set. It defines a partition of the set consisting of one-element subsets: R = {(x,x) \Χ(ΞΕ], V = {{X}\ xeE}. 2) The relation that associates any two digits whenever they are both even or both odd is an equivalence relation introduced by the partition V - { { 0 , 2 , 4 , 6 , 8 } , {1,3,5,7,9}} of the set F = {0,1,2,3,4,5,6,7,8,9}. 3) Consider a set of statements associated with subsets of a set U so that the implication A Β means A C Β for the corresponding sets. Then the logical equivalence A Β of the statements A and Β means A = Β for the corresponding sets. The equality of sets is an equivalence relation on the class V = V(U) of all subsets of U. 3. A less trivial example of an equivalence relation uses the definition of cardinality, which is a formal generalization of the intuitive notion of the number of elements in a set. Let V = V(U) be the class of all subsets of a set U. By we denote the relation on V induced by an isomorphism of two sets: for Χ, Υ € V, X Y means that X and Y are isomorphic (i.e., there exists a one-to-one map from X onto Y). The relation is an equivalence relation. Indeed, it is reflexive (the identity map from X onto itself is an isomorphism), transitive (the composition of isomorphisms from X onto Y and from Y onto Ζ is an isomorphism), and symmetric (the inverse of any isomorphism from X onto Y is an isomorphism from Y onto X). Thus, V can be partitioned into a collection of pairwise disjoint nonempty classes, each consisting of sets isomorphic to each other. Isomorphic sets are said to be equipollent, and classes of sets isomorphic to each other are called cardinals. For example, the class 0 = {0}, whose only member is the empty set, and the class 1 = {{χ} | χ € U} of all singletons {ζ} € V are cardinals. For a set X, the class of all sets isomorphic to X is called the power of X or the cardinal number of X, and is denoted by |X|. This class depends on the given universal set U whose subsets are elements of To emphasize this dependence, the notation \X\u can be used instead of By Dedekind's definition, any set X that is not isomorphic to any of its

Chapter 1. Set theory

17

proper subsets is said to be finite . If X is finite, we write |X| < oo. Otherwise, X is said to be infinite (|X| = oo). Obviously, every subset of a finite set is finite. Operations defined for cardinal numbers are described in detail in Kuratowski and Mostowski (1970, Chapter 5), and Bourbaki (1965, Chapter 3). Exercise. Prove that if .A is a finite set and Β is an infinite set, then \AUB\ = \B\. The following theorem provides a criterion for two sets to be equipollent. The Schröder-Bernstein Theorem. Sets A and Β are equipollent if and only if there are injections of the set A into the set Β and of Β into A. The proof can be found in Kuratowski and Mostowski (1970, (Chapter 1, Corollary 4.4). Remark. In naive set theory, which is employed here, it is assumed that the introduction of the universal set helps to avoid some paradoxes associated with the set of all sets (Kuratowski and Mostowski, 1970, Chapter 2, Section 3). The existence of the class of all subsets of a given set is postulated by the Axiom of power set (Kuratowski and Mostowski, 1970, Chapter 2, Section 2). For this reason, in the present book, by the term a class of sets we usually mean a class of subsets of a certain universal set. These notions are assumed to be self-evident no formal definition is provided for them.

1.3.3.

Order

Any reflexive and transitive relation is called an order. Most orders considered in practice possess some additional properties. For example, any symmetric order is an equivalence relation. The union of any transitive relation and the equality relation is an order. Order is not assumed to be defined on the entire set and therefore is also called partial order. 1. Let R be an order on a set E. We will write y Χ χ instead of (x, y) € R and say that the element y succeeds to the element x. An important type of order is an antisymmetric order. Moreover, the terms order and partial order usually imply antisymmetry. A relation that is only reflexive and transitive is called a preorder. For antisymmetric orders,

18

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

we write y > χ instead of y b % and say that y is greater than x. By the definition of an antisymmetric relation, χ > y and y > χ implies y = x. For an order R on E, its inverse R~1 is an order on E. We will write y ^ χ instead of (a;, y) G i ? - 1 and say that the element y precedes the element x. If R is antisymmetric, then R~1 is also antisymmetric. For antisymmetric orders, we write y < χ instead of y ^ χ and say that y is less than x. The negation of a relation statement will be denoted by a slash mark across the relation sign. An order can be viewed as "direct" and denoted by >z in one case, whereas in other contexts it may be convenient to view the same order as inverse and denote it by since these terms and designations are relative. If y y χ and y ^ x, we write y >- χ and say that y strictly succeeds to x. liy> χ and y % x, we write y > χ and say that y is strictly greater than x. Note that the condition y ^ χ is equivalent to y φ χ by antisymmetry. The designations y -< x, y < x, and the terms strictly precede and strictly less are defined similarly. Any elements χ and y such that y >z χ οτ χ >z y are said to be comparable. An order such that any two elements of the ordered set in question are comparable is called a linear order or a linear order. For a total antisymmetric order, the trichotomy law holds: for any y and x, either y > x, or y = x, or y < x. Example. For any set, the set of cardinals of its subsets is linearly ordered (see Kuratowski and Mostowski, 1970, Chapter 1, Corollary 5.5). Any two elements χ and y of a partially ordered set such that y Χ χ and χ y y are called order-equivalent or simply equivalent. It is easy to verify that this relation is reflexive, transitive, and symmetric. This equivalence will be denoted by =. Exercise. Prove that Ξ is an equivalence relation. Any set Ε taken together with an order >; on it, i.e., the pair (E, >;) is an ordered set. It is often denoted simply by Ε whenever the meaning is clear from the context. Examples. The usual order on the set of digits can be written as 0 < 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8 < 9 . It is a total antisymmetric order.

Chapter 1. Set theory

19

The set of all ordered pairs of digits can be ordered with respect to their first elements. For any digits o, 6, x, y, let ab < xy mean that a < x. This order is not antisymmetric. For example, 12 < 13 and 13 < 12 although 12 φ 13. The inclusion relation D is an order on the class V = V(U) of all subsets of a set U. This order is antisymmetric, but it is not total if there exist at least two distinct elements in U. 2. Let X be a subset of a set Ε and >z be an order on E. If b e Ε succeeds to every χ € X, then b is said to succeed to the set X, which is denoted by b >z X. If an element a € Ε precedes every χ ε ί , then a is said to precede the set X, which is denoted by a ^ X. Any ordered set Ε such that for any x,y € Ε there is a ζ Ε Ε that succeeds to χ and y is called a directed set. Directed sets are sometimes called nets (because of the graphical representation of elements of directed sets in the form of net points). We emphasize that the order that defines a directed set is not necessarily antisymmetric. 3. Assume that an order on Ε is antisymmetric. For b Ε Ε and X C Ε such that b succeeds to X, we say that b is an upper bound of X, or X is majorized by b, or 6 is a majorant of X. If an element α precedes X, then α is called an upper bound of X, or a minorant of X. An upper bound of X that is contained within X is called the greatest element of X and denoted by m a x l . A lower bound of X that is contained within X is called the least element of X and denoted by min X. The least element of a set is sometimes called its initial element, and the greatest element is called its last or terminal element. The least upper bound of X is called the supremum of X and is denoted by sup X. The greatest lower bound of X is called the infimum of X and is denoted by inf X. The supremum and infimum of X do not necessarily belong to X. The notions of infimum and supremum are extensions of the notions of the least element and the greatest element. An element that is not smaller than any other element in Ε is called a maximal element. An element that is not greater than any other element in Ε is called a minimal element. These notions are extensions of the notions of the smallest and the greatest elements of an ordered set. The greatest and the least elements of an ordered set are necessarily unique (if they exist). This is not the case for maximal and minimal elements. For any two comparable elements χ and y, however, the greatest

20

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

element and the least element of the set {x, y} do coincide with its maximal and minimal elements, respectively. The least of them is denoted by min{:r, y} or χ Λ y, and the greatest by max{x, y} or χ V y. As these notions are sometimes easy to confuse, special care must be taken when using them for partial orders. Examples. Consider the class V = V(U) of all subsets of a set U and the order relation D for V. Clearly, U = max Ρ and 0 = min "P. It follows from the definitions that sup{X, Y} = XUY and inf{X, Y} = Χ Π Y for any X,Y 6 V. Furthermore, for any class C C V, supC = UC and inf C = ΠC. The set Ε = {00,01,..., 09,10,20,..., 90} with order relation 00 < 01 < • · · < 09; 10 < 20 < · · · < 90 has two maximal and two minimal elements. 4. Let Ε and F be ordered sets. We will take the liberty of using the same designation >; for both orders. A map / :E-+F such that f(y) y f{x) whenever y y χ (x,y £ E) is called monotonic. A monotonic map is also said to be order-preserving. If f(y) ^ /(:r) whenever y >z x (x,y 6 E), then / is called antimonotonic. Antimonotonic maps reverse the order. Monotonic and antimonotonic maps defined on ordered sets with antisymmetric orders are called increasing and decreasing, respectively. Let Ε and F be ordered sets with antisymmetric orders denoted by > in both cases. A map / : Ε —> F such that f(y) > f(x) whenever y > χ (y,x £ E) is said to be strictly increasing. A map / : Ε —• F such that f(y) < f(x) whenever y > χ (y,x € E) is said to be strictly decreasing. If Ε and F are ordered sets, a strictly increasing one-to-one map / from Ε to F such that f"1 is strictly increasing is called an embedding from Ε into F. If f(E) = F, then / is called an isomorphism from Ε onto F and the ordered sets Ε and F are called isomorphic. These definitions of embedding and isomorphism are extensions of similar definitions for sets in general, the added requirement being the preservation of order. If there exists an isomorphism from an ordered set Ε onto an ordered set F, then the corresponding orders are called equivalent. Examples. For Ε = {α, β, 7, δ, ε, ζ, η, θ, L, κ} and F = {0,1,2,3,4,5,6, 7,8,9}, we define the orders α 1 and \Y\ > 2, then \YX\ > \X\. For example, for Υ = {0,1}, the set of all indicators on X is equipotent to the class 2X — V{X) of all subsets of X, and \2X\ = \V(X)\ > \X\ by the Cantor theorem (Kuratowski and Mostowski, 1970, Chapter 5, Section 3, Theorem 2; Bourbaki, 1965, Chapter 3, Section 3, Theorem 2). It is easy to verify that if the orders on A and Β are antisymmetric, then so are the coordinatewise and the lexicographic orders on Α χ Β. In this case, the condition χ ξ α in the definition of the lexicographic order can be replaced with the condition χ = α. If the orders on A and Β are total, then the same is true for the lexicographic order on A χ J3, but not necessarily for the coordinatewise order, which was illustrated in the previous example. It is clear that the cartesian product and the lexicographic product of directed sets is a directed set. 6. Let V be a class of pairwise disjoint ordered sets. It is possible to introduce an order on WD based on the orders on sets in V and an antisymmetric order on V. For χ £ X and y € V, where Χ,Υ Ε T>, we put y y χ if (1) Υ > X in V or (2) Υ = X and y y χ in X. The union U V ordered this way is called the order sum of the sets of the class V. It is denoted by J2 V. Example. Let V be the class containing the sets X = F = { 0 , 1 , . . . , 9}, y = (F\ {0}) χ F= {10,11,..., 99}, and Ζ —Υ x X = {100,101,..., 999}. Let the orders on X, Y, and Ζ be defined by the inequalities 0 < 1 < • · · < 9, 10 < 11 < · · · < 99, and 100 < 101 < · · · < 999, respectively. Let the order on T> be defined by the inequalities X < Υ < Ζ. The union UV = XUYUZ becomes the order sum ΣΤ> = Χ + Υ + Ζ if we introduce two more inequalities: 9 < 10 and 99 < 100. It is easy to verify that if the orders on the component sets are antisymmetric, then so is the order on the union. Furthermore, if V and each set in it are linearly ordered, then so is the union.

Chapter 1. Set theory

23

Exercise. Prove the foregoing statements.

Remark. The order on UT> used in the definition of the order sum Σ ^ is related to the lexicographic order. Indeed, for any two elements, we first compare the sets that contain these elements. Then, if these sets are equal, the elements themselves are compared. That is, we use the lexicographic order for pairs of the form (X, x) € VxLiV such that X € V and χ € X. The pairs (X, y) such that X € V and y g X are not taken into consideration.

1.4.

INDUCTION

In this section, we prove several theorems for specific ordered sets. These theorems are often used in mathematical argument. All the orders considered in this section are assumed to be antisymmetric. 1.4.1.

Well-ordered sets

An ordered set Ε with order > is said to be well ordered if every nonempty subset of Ε has a least element. The order > is called a well-order relation. Obviously, every well-order is a linear order. If > is a well-order, its inverse < is not necessarily a well-order. In particular, if there is no greatest element in a well-ordered set Ε with wellorder >, then < is not a well-order. The empty with empty order relation is a well-ordered set. 1. Every two elements a,b G Ε with linear order > such that a < b define the following intervals in Ε with endpoints α and b: ]α, 6[ = {χ I a < χ < 6},

[a,b[ = {χ \ α < χ < b},

]a, 6] = {χ I a < χ < 6},

[a,ft]= {χ | a < χ < b}.

Intervals of the form [α, 6] are also called closed intervals, and intervals of the form ]a, are called open intervals or segments. An interval of either of these types with endpoints α and b will be denoted by |a, b\. The elements a and b are also called the left endpoint and the right endpoint of the interval MlNote that if α = b, then the interval [a, a] = {a} consists of a single point, while ]o, a[ = [a, a[ = ]a, a] = 0 . Intervals of the form ]a, b[ may be empty even if α φ b.

24

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed For any α € E , the following intervals in Ε with endpoint α are defined: ]α, —>[ = {χ I a < χ},

[a, —>[ = {χ \ α < χ}.

An interval of either of these types with endpoint α will be denoted by |a, — Note that if Ε has a greatest element, b = max Ε1, then |a, —>1 is the same as [a, b] or ]a, 6]. Similarly, we introduce intervals of the form ]. Let α € Ε. Then L = {a} is a chain in E. By the Hausdorff maximality theorem, there exists a maximal chain Μ in Ε such that M D L . Assume that every chain in Ε has an upper bound, and let b 6 Ε be an upper bound for the chain M. Then α < b. It follows that the element b is maximal. Indeed, suppose b < c for some c e E. Then χ < b < c for all χ e Μ and Ν - Μ U {c} is a chain in E, which is a contradiction because Μ is a maximal chain. • Remark. Axiom of choice, the Hausdorff maximality theorem, and Zorn's lemma are equivalent. A brief proof of this statement is given in Kelley (1975).

1.5.

NATURAL NUMBERS

Natural numbers provide a way to count and specify the order of elements in well-ordered sets. The role of natural numbers in mathematics is described in the famous phrase of Leopold Kronecker: "God made the integers, and all the rest is the work of man". The natural numbers are formally defined by a system of axioms. For example, see Kuratowski and Mostowski (1970), Chapter 3, where a brief description of the natural numbers in terms of naive set theory is provided. For details, see also Savel'ev (1969), Introduction to Chapter 2. 1.5.1.

Decimal natural numbers

The most common representation of natural numbers is the one that uses the set of digits F = { 0 , 1 , . . . , , 9}. We will also consider the set FQ = { 1 , 2 , . . . , 9} of nonzero digits. The set Ν of decimal natural numbers will be defined by induction as a discrete infinite set. First, we specify an infinite discrete class of sets using the sets F and FOThe initial element of this class is the set α — Fo. For an element χ that is already defined, its immediate successor x+ is defined as the cartesian product of Χ and F: X+ = Χ χ F (Χ G Ν). In particular, we have a+ = F 0 χ F = {10,11,..., 19; 20,21,..., 29;... 90,91,..., 99}, (a+)+

= F

0

X F X F ,

((a+)+)+ =

F0

x

F

χ

F

χ

F.

28

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

Thus, the initial element α and the rule of obtaining x+ from χ defines the well-ordered class Λί=

{Fo,Fo χ F,F0 x F χ F,...

}.

Clearly, this class consists of pairwise disjoint sets. Every set in Af can be ordered with the lexicographic order. The order sum

represents ordered set of decimal natural numbers. 1.5.2.

The isomorphism theorem

Aside from the decimal representation, there are many other representations of natural numbers. This is justified by the following theorem. Theorem. For any two infinite discrete sets A and B, there exists a unique monotonic isomorphism from A onto B. Proof. 1) We now prove the existence part. Assume α = min A, b = min ß and consider the correspondence ψ : A —• Β such that φ(α) = b and ψ{χ+) = φ{χ)+ (x & A). Let X be the set of all χ Ε A such that φ is defined and is strictly monotone on the interval [a, x\. Put ψ(χ) = Y. It follows from the definition of φ that α Ε X, b € Y. Furthermore, for any χ Ε A and y Ε Β, if χ Ε X and y Ε Y, then (χ+) Ε Χ and (y + ) Ε Υ. By induction, Χ = Α, Υ = Β, and ψ is the required isomorphism. 2) We now prove the uniqueness of ψ. Let t/> be a monotonic isomorphism from A onto B. We have ψ(α) = b = φ(α) and the equality ψ(χ) = (Η, +) have the following properties: Φ{χ +

Ν)

= φ{χ)φ{Ν),

IP(XY)

= Φ(Χ)

+ Φ{Ν)

(x,yEG).

All definitions for arbitrary maps apply to homomorphisms: one-to-one homomorphisms are called embeddings, surjective embeddings are called isomorphisms, and the isomorphisms of a semigroup onto itself are called automorphisms. For isomorphic semigroups, we write (G, g) ~ (Η, h) or G ~ H, meaning that there exists an isomorphism ψ : G —» Η and its inverse φ~ι : Η —> G.

Chapter 2. Algebra

35

Exercise. Prove that the map inverse to an isomorphism from a semigroup to a semigroup is an isomorphism. Examples.

1) The map ψ{η) = 2n is an isomorphism from ( N , + )

onto (2N,+). 2) The map φ(Χ) — X' is an isomorphism from (V, U) onto (V, Π) and of (V, n) onto (V, U). This follows from the property of duality

(XuY)' = Χ'ΠΥ', {xnY)' = x'uY' (x,Yer). The isomorphism φ coincides with the isomorphism φ~ι because the operation of taking the complement of a set is involutive. 7. Any semigroup (H,h) is isomorphic to a subsemigroup (T(H),·) of the semigroup (!F(H), ·) of transformations of Η (see Clifford and Preston, 1967). The elements of T{H) are the left-shift operators defined for any χ € Η by the formula τχ(ζ) = h(x, ζ) (ζ € H). By the associativity of h,

τχ(τν(ζ)) = h(x,h(y,z)) = h(h(x,y),z) = Th{x,y)(z) ( and therefore (T(H),·) is a semigroup. Clearly, T(H) C T(H). The preceding equalities for the operators τ χ , r y , and imply rh^x y ) = τ χ • r y (x, y e If). Consequently, the map τ : Η —> Τ (Η) is a homomorphism from (H, h) onto (T(H), •) or into {F(H), ·). Exercise. Determine the conditions under which this homomorphism becomes an isomorphism. Example. The operator representations of (N, + ) and (N, ·) are isomorphic to these semigroups. For this reason, natural numbers can be viewed as operators. We could consider the right-shift operators instead of the left-shift operators. For semigroups with commutative operation, they coincide and are therefore called simply shift operators. 8. For any set E, the set ΊΖ — 7t(E) of all relations on Ε with the operation ο of composition of relations is a semigroup. We will use multiplicative notation and write TS (or ST) instead of Γ ο 5 and Rx (or xR) instead of R(x) for R,S,Te TZ. Whether to use the left-side or the right-side notation is a matter of convention. The variants TS and Rx are more common.

36

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

Note that 1Z = V(E χ Ε), i.e., every relation R is a subset of Ε χ Ε. A relation R can be viewed as a map from Ε to V(E) such that R(x) = {y \ (x, y) e TZ] for χ 6 Ε, so that TZ = V{E)). We now verify that the composition of relations is associative. Let R,S,T 6 TZ and χ € Ε. Prom the definition of the composition of correspondences, it follows that ((TS)R)(x)

= (TS)(R(x))

= T(S(R(x)))

=

T((SR)(x)),

and thus ( T S ) R = T(SR). For this reason, (TZ, ο) = (1Z, ·) is a semigroup. The semigroup (Τ, ο) = ·) of transformations of Ε can be embedded into the semigroup (7Z, ·) and is therefore a subsemigroup of (TZ, ·). The identity transformation I is the identity element in (TZ, ·). The empty relation 0 is the zero element in (TZ, ·). The semigroup (7Z, ·) is noncommutative since ST = TS is not necessarily true. Example. Let Ε = Β = {0,1}, TZ(E) = TZ(M) = P(B χ B) = ^{00,01, 10,11} (24 = 16 relations in total), S = {00,01}, Τ = {00,10}. Then ST = {00} Φ {00,01,10,11} = TS. Such compositions can be illustrated by the following diagrams:

1

1

1

1

1

1

1

1 1

"1

The above example also illustrates the duality of the functional notation (ST)(x) = S(T(x)) and the operator notation χ (ST) = (xS)T. Note that S and Τ are inverses of each other as correspondences: (0,0) (0,0), (0,1) (1,0). In the semigroup (TZ(E), ·), however, S and Τ are not inverses of each other. The algebraic inverse of an element may not coincide with its inverse as a correspondence. Correspondences and, particularly, relations are sometimes called multivalued functions or multivalued abstract operators. Their values are sets. 9. The following statements hold for compositions of homomorphisms. Proposition. The composition of homomorphisms of semigroups is a homomorphism.

37

Chapter 2. Algebra

Proof. Let F, G, and Η be semigroups, α : F —• G and β : G Η be homomorphisms, and 7 = βα : F —• Η be the composition of α and β. It is required to prove that 7 is a homomorphism. In additive notation, we have η(χ + y) = β(α(χ + y)) = β(α(χ) + a(y)) = ß(a(x)) + ß(a{y)) = 7(x) + for all x,y Ε F. Consequently, 7 is a homomorphism.

7(y)

•

For sets, the composition of isomorphisms is an isomorphism (see Section 1.2.3). Corollary. The composition of isomorphisms of semigroups is an isomorphism.

Exercises. Prove the following statements for relations R, S, Τ G 1Z(E): 1) R C S =» RT C ST, TR C TS. 2) RS C

RS — SR if R and S are symmetric.

3) If S i s the inverse of S as an automorphism of E, then S - 1 is the algebraic inverse of S as an element of the semigroup (71(E), •).

2.1.2.

Groups

A semigroup is called a group if it has a neutral element and every element of this semigroup has an inverse. Groups are the main subject of Chapters 2 and 7 in the monograph van der Waerden (1991). Different versions of the definition of a group are discussed in Clifford and Preston (1967). 1. All general definitions and propositions for semigroups can be extended to groups. A subsemigroup of a group is its subgroup if it is itself a group. The neutral element of a group belongs to its every subgroup. Indeed, for any element of a subgroup, its inverse also belongs to it and the group operation applied to them yields the neutral element.

38

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

Examples. 1) The semigroups (V, U) and (V, Π) are not groups: X U Y = 0 only if X = Y = 0 , and Χ ΠΥ = U only if X = Y = U (V = P(U), Όφζ). 2) The semigroup (71, ·) of all transformations of the universal set U and their compositions is not a group. At the same time, the semigroup (.A, ·) of all automorphisms of U is a group since for any R € A its inverse element in the semigroup is its inverse automorphism R~1 € A. 3) The additive semigroup (B, +) with Β = {0,1} is a group. We have 0 + 0 = 0, 1 + 1 = 0. We call (B, +) the binary group. The multiplicative semigroup (B, ·) is not a group, whereas the semigroup (Β \ {0}, ·) = ({1}, ·) is a group. 2. A classical example of an Abelian group is the additive group of integers (Z, +). It is obtained from the semigroup (N, +) by adding zero and the inverse elements. Every natural number η is associated with the negative integer number —n. The set of all negative integers is denoted by —N. The set of integers is the union Ζ = (—Ν) U {0} U N. The operation of addition in Ζ is defined in the usual way. The set Ζ with addition, written (Z, +), is an Abelian group. The order on —Ν is determined by the order on N: —m < —n if m · 2 = 0, then a(x) = a(y) => 0 = a(x) - a(y) -- a(x - y) =>· x — y = 0=t>xz=y and therefore a is an embedding. • Exercise. Prove this lemma for a homomorphism β : (F, ·) —> (G, ·). 6. We now prove a lemma on a connection between homomorphisms and normal subgroups. Lemma. The kernel of a homomorphism is a normal subgroup. Proof. Let α : (F, +) ->• (G, + ) , Ν = Ker a, u Ε Ν, ν € Ν, χ G G. 1) Since a(u + v) = a(u) + α(υ) = 0 + 0 = 0, we have u + ν Ε Ν. Furthermore, α(0) = 0 implies 0 Ε Ν, and a(—u) = —a(u) = —0 = 0 implies — u € N. It follows that (TV, + ) is a subgroup of F. 2) Let x + u = y be an arbitrary element of x + N. Then u = —x + y and 0 - a(u) — —a(x) + a(y). Hence a(x) = a(y). We have 0 = a(y) — a(x) = a(y — x) and y — χ = ν € Ν. Then y = v + xeN + x and χ + Ν C Ν + χ. The inclusion x + N D Ν + χ is established in a similar fashion. Thus, χ + Ν = Ν + χ and iV is a normal subgroup. • Exercise. Prove that every normal subgroup is a kernel of a homomorphism. 7. Every normal subgroup Ν of a group G defines an equivalence relation on G that is consistent with the group operation. This provides a way to define a new group G/N that is homomorphic to G. We will examine this kind of groups in detail because of its importance. Lemma A . The relation χ ~ y x — y G Ν (x,y € G) is an equivalence on (G, +) consistent with the addition operation. Proof. 1) We have 0 = χ — χ~ y y — χ = — (χ — y) Ε Ν =Φ· y ~ χ and x~y,y~z=>x — y,y — zEN=>x — z = (x — y) + (y — z)E Ν =>· χ ~ ζ. It follows that ~ is an equivalence relation. 2) Let a,b,x,y Ε G and χ ~ ν Ε N, and ν — a = —a + w for Then (x + y) - (a + b) = x + y-b and χ + y ~ a + b. This means addition operation. •

a, y ~ b. We set χ — a = u Ε Ν, y — b = some w 6 Ν (since Ν + (—α) = (—ο) + Ν). — a = x + v — a = x — a+ w = u+ w Ε Ν that the relation ~ is consistent with the

42

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

Lemma B. The relation ~ deßned by the condition χ ~ y -ΦΦ- xy~l € Ν (χ, y € G) is an equivalence on (G, ·) consistent with the multiplication operation. Exercise. Prove Lemma Β using arguments similar to those used in the proof of Lemma A. We introduce the following notation for the sake of convenience: α = α + Ν (α 6 (G,+)),

α = αΝ (α € (G, ·)),

G = G/N.

From the definitions, it follows that elements α and b of the group G are equal if and only if the corresponding elements α and b of the class G are equal, i.e., α ~ b α = b. Corollary A. For the group (G, +), the class G = G/N of sets α = α+Ν (α Ε G) is a partition of the set G. An addition operation in G is deßned as follows: a + b = α + b (a, b 6 G) This operation is associative and there exist a neutral element and inverse elements with respect to it. Proof. 1) From Lemma A and the theorem of Section 1.3.2 (1), it follows that G is a partition of G. 2) The addition operation in G is well defined because the equivalence relation ~ is consistent with addition in (G, +). Indeed, χ = ä, y = b χ + y = a + b χ + y = a + b (a,b,x,y Ε G). 3) The associativity of addition in (G, +) ensures the associativity of addition in G: ö + (b + c) = ä + (b 4- c) = α + (b + c) = (a + b) + c = (a + b) + c= (ü + b) + c (a,b,c€ G). Since ö + Ö = α + 0 = ö = 0 + α = Ö + ά (a € G), it follows that Ö is the neutral element with respect to addition in G. Since ö + (—a) = a + (—a) = Ö = (—a) + a = (—a) + a, we have ^ ä = —ä (a € G). • Corollary B. For the group (G, +), the class G = G/N of sets α = αΝ (α € G) is a partition of the set G. A multiplication operation in G is defined as follows: ab = ab (a, b € G) This operation is associative and there exist a neutral element and inverse elements with respect to it.

43

Chapter 2. Algebra

Exercise. Prove Corollary Β using arguments similar to those used in the proof of Corollary A. Corollaries A and Β imply that (G/N, +) and ( G / N , ·) are groups. They are called factor groups of the groups (G, +) and (G, ·) over the normal subgroups (N, +) and (N, ·), respectively. Examples. 1) Let (G,+) = (Z,+) and (N,+) = (2Z,+). Then the factor group {G/N, +) = (Z/2Z, +) is isomorphic to the binary group (B, +). 2) For every normal subgroup F of a group G there exists a factor group G/F. In particular, for the subgroup F = {e} containing only the neutral element, the factor group Gj{e} is isomorphic to G. The factor group G/G is isomorphic to the trivial group {e}. 3) For every homomorphism φ : F —> G, there exists a factor group F/Ker φ. An example of such a group is (Z/2Z,+).

Remark. From the definitions, it follows that the factor sets that make up the factor group G/N are equipollent to the normal subgroup Ν. Assume that the sets G, N, and G consist of p, n, and p elements, respectively. Then ρ = ρ • η and ρ = ρ/η. 8. For the group G and the factor group G, there is a natural homomorphism φ : χ —> χ (χ G G). The following equalities hold in the additive and the multiplicative cases, respectively: φ(χ + y) = X + y = x + y = φ(χ) + · G/N, the inverse correspondence ψ : G —> G, and the composition 7 = φφ~1 : G —• G/N.

G/N The correspondence 7 is injective. Indeed, by assumption, for every α G G there is an α € G such that φ(α) = α. If χ Ε G and φ(χ) = ä, then φ(χ) = φ(α), χ ~ α and φ(χ) = ψ{α). Hence, 7(0) = {&{%) | ψ{χ) — α} = φ(α).

45

Chapter 2. Algebra

We now prove that 7 is a homomorphism. Let ö = φ(α) and b = G be the isomorphism inverse to 7. Then, by the proposition of Subsection 2.1.1, part 10, the composition ψ = η~ιφ : G —• G of homomorphisms φ and 7 " 1 is a homomorphism. Since φ maps G onto G/N and 7 - 1 maps G/N onto G, X ψ maps G onto G. Finally, ker = (Ö) = ^ - ^ ( Ö ) ) = ψ~1(Ν) = Ν. • The homomorphism theorem has a number of useful corollaries. 10. Let G be a group, F be its subgroup, and Ν be its normal subgroup. We will use multiplicative notation and write AB = {xy | χ € A, y G B} for A, Β CG. Theorem. F/{F η Ν) ~ FN/N. Proof. Prom the definitions, it follows that F(~)N and FN are subgroups of G, F C FN, Ν C FN, and F Π Ν and Ν are normal subgroups of F and FN. It follows that the factor groups in the assertion of the theorem are well defined. The following diagram illustrates the proof. The natural homomorphism from G onto the factor group G/N is denoted by 7, while a and β are its restrictions to F and FN, respectively. The maps 1, j, ϊ, j are identity embeddings, which are homomorphisms. From the definition of 7, it follows that ker 7 = N. G

- G

Μ

J

=

_L

FN•t F

= G/ Ker 7

ί

- FN ~ FN/ Ker β = a

-t t

G/N

F/Kera

FN/N

= F/(F Π Ν)

By the homomorphism theorem, F ~ F/ker α, FN ~ FN/ker/3. We have ker/3 = ^"^Ö) - ( r ^ j ) - 1 ^ ) = J _ 1 7 _ 1 J(Ö) = f 1 ! " 1 ® = 3 ~ \ N ) = FNDN = N, ker a = a _ 1 ( 0 ) = (t- 1 /?»)- 1 ^) = = ι^β-^Ο) = ι~1(Ν) = F Π Ν. This proves the horizontal equalities in the diagram.

46

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

From the definitions, it follows that j~l(F) = FN. Therefore, FN = j(FN) = 7 ( 7 _ 1 ( F ) ) = F, which implies that Ί : F YN is an identity isomorphism and F/(F Π Ν) ~ F = FN ~ FN/N. • This corollary of the homomorphism theorem is called the first isomorphism theorem. Exercise. Prove that FN is a subgroup and FC\N is a normal subgroup of G.

Μ

1 1 . Let G be a group, Ν and Μ be its normal subgroups such that CN. Theorem. G/N ~

(G/M)/(N/M).

Proof. It is easy to see that Μ is a normal subgroup of N . The proof is illustrated by the following diagram: G

"G = G/N ~

(G/M)/(N/M)

a G = G/M By 7 and β we mean the natural homomorphisms onto the corresponding factor groups, while α = ηβ~ χ is the composition of the correspondence βand the homomorphism 7. We now prove that α : G —> G is a homomorphism. Let χ G G and ß{x) = f G G. Then β~ι(χ) = xM and a ( f ) = η(β~ι(χ)) = η{χΜ) - xM · Ν = χ · Μ Ν = χΝ = η {χ) = χ Ε G (since Μ C Ν and 1 € Μ , we have Μ Ν = Ν). The correspondence α is injective. Let x,y € G and χ = β (χ), y = ß{y). We obtain xy = ß(x)ß(y) = ß(xy) = xy and a(xy) = a(xy) = 7 ( ß ~ l ( x y ) ) xyM) = 7 ( x M · yM) = 1 _1 7 ( x M ) • j(yM) = η(β~ {χ)) · η(/3 (y)) = α(χ)α(y). Hence, a is a homomorphism from G into G. By the homomorphism theorem, G ~ G/ker a. We conclude that kera - a _ 1 (Ö) = (7/3" 1 )- 1 (Ö) = ^(7 - 1 (JV)) = ß(N) = {ß(x) I χ G N } = {x G Μ I χ € Ν } = N/M. • The foregoing corollary from the homomorphism theorem is called the second isomorphism theorem. It describes the successive factorization of a group. Such factorization can be useful for constructing general solutions to equations.

Chapter 2. Algebra

47

Example. Z/2Z ~ (Z/4Z)/(2Z/4Z). Exercise. Use the isomorphism theorems to prove that (.FM/M)/((F nN)/(FHM))

~

FN/N.

Conversely, prove that this equality implies the first and the second isomorphism theorems for Μ = {1} and F = G, respectively. We now formulate a useful proposition that follows from the theorems proved above. Let F, G, and Η be groups and let β : G —> F and 7 : G Η be surjective homomorphisms. Proposition. The inclusion ker β C ker 7 is true if and only if there exists a unique homomorphism α : F —> Η such that αβ = η. Proof. Let Μ = ker β and Ν = ker 7. By the homomorphism theorem, there exist isomorphisms ψ : F —> G/M and ψ : Η —• G/N. If Μ C N, then there is a natural homomorphism G/M —> (G/M)/(N/M) and, by the second isomorphism theorem, (G/M)/(N/M) ~ G/N. Then there exists a homomorphism ά from G/H onto G/N. The required homomorphism is α = φ~ιδίψ. If there is a homomorphism α such that aß = j, then for χ € G we have β{χ) = 0 η(χ) — α{β{χ)) = α(0) - = 0 and ker/? C ker7. The uniqueness of α follows from the equalities ο. = ψαφ~ι and α = ψ-1 αφ. • The proof is illustrated by the following diagram (ß : G —> G/M, G —> G/N are natural homomorphisms).

7 :

Φ a

G

a

Ψ 12. Any group G is isomorphic to the group Τ = T(G) of left-shift operators. In additive and multiplicative notations, the shift ra by an element a € G is defined as follows: τα(χ) = a + χ, τα(χ) = αχ

(χ e G).

48

Μ. Lavrent'ev and L. Savel'ev. Operator theory and ill-posed

...

In part 7 of Subsection 2.1.1, we proved that (T, + ) and (T, ·) are subgroups. They are groups whenever G is a group. The shift by the neutral element is the neutral shift. For every shift τ α , its inverse is the shift by the element inverse to a: TQ(X)

τα(τ-α(χ))

= 0+

Χ

=

Χ,

τ\(χ) = 1 ·

= a + (-a + χ) = χ,

Χ

= x;

τ α (τ 0 -ι (χ)) = ο(α _ 1 χ) = χ;

τ α τ_ α = r 0 = id,

τ α τ α -1 = id.

The same is true for right-shift operators. For Abelian groups, the notions of right shift and left shift coincide. 13. Let Ε be a set, (F, + ) be an Abelian group, and let (J7, + ) be an Abelian group such that Τ = Τ(Ε, F) is the set of all maps from Ε to F and the addition operation for /, g G Τ is defined as follows: V + g)(x) = f(x) + g(.x)

(x€E).

The neutral element in Τ is the constant 0. For / 6 Τ , its inverse is —/ G Τ such that (—f)(x) = — f(x) for all χ G E. The addition operation on Τ is commutative because so is the addition operation on F . If (Ε, + ) is a semigroup, then every element α Ε Ε determines the difference operator Aa : f —> Aaf (/ G J7), where a function Aaf is defined by the formula Aaf(x)

(x G E).

= f(a + x)-f(x)

The operator Δ α is additive, i.e., it is a homomorphism from the group (F, + ) into itself: Δα(/ + g)(x) - (/(α + X) + g(a + x)) - (f(x) + g(x)) = (/(a + x) - f(x)) + (g(a + x) - g(x)) = Aaf(x)

+ Aag(x)

(f,g€f).

It follows from the definitions that Δ„/ = / τ β - /

(/€ Τ).

14. Let (£?,+) = (N,+), (F,+) = (Z,+), JF = ^(N,Z), and a = 1. Then, for the difference operator Δ = Δι and a function / G T, the function Δ / G Τ is defined by the formula Af(x) In particular, Af(x)

= f(x + l)-f(x)

(x G N).

= 1 for f(x) = χ and Af(x)

— 2x + 1 for f(x) = x2.

49

Chapter 2. Algebra Let g Έ T. We now solve the difference equation Af = g,

Af(x)

= g(x)

(x G N).

Prom the definitions, it follows that /(2) = /(1) + g(l) and f(x + 1) = f(x) + g(x) for ι ξ Ν , By induction, the function / is defined by /(1) and the equality / ( s + l) = / ( l ) + Ε 9(n) ( z € N ) . l F (| supp/| < oo).

51

Chapter 2. Algebra

Example 2. Consider an infinite field K, the nth power function sn : Κ -* Κ with values sn(£) = ξη (ξ € Κ), and the additive group G(n) = Ks n = {as n I α € Κ} generated by sn such that (a + ß)sn = a s " + ßsn (α, β € Κ). The additive group of polynomials G = .M(K) is the direct sum of power groups G(n) = Ks n (η = 0,1,2,...). The group of polynomials ΛΊ(Κ) is a subgroup of the additive group of functions .F(K, K). For a polynomial ρ : Κ —» Κ with values ρ(ξ) = öo + αιξ + · · · + αηξη (αη φ 0), its coefficients on (0 < i < η) are uniquely determined if the field Κ is infinite. An additive Abelian group G is the direct sum of a finite collection of subgroups G(i), written G = Σ G(i), if for any χ € G there exists a unique collection χ = (χ(«)) € Π ^ ( ^ ) s u c ^ that χ = Σ χ ( ί ) · For the direct sum of groups and subgroups, there following correspondence is an isomorphism: φ :l[G(i)-+Gt

φ(χ)=χ

=Σ

χ

^

(x =(«( 0 and q, there exists a natural number η such that np > q. Proof. Prom the preceding lemma, it follows that there exists a natural number η > r = q/p. This inequality is equivalent to the desired one. • Obviously, the inequality np > q in the theorem can be replaced by the strict inequality np > q. In view of Archimedes' theorem and the consistency of the order with the operations, (Q, + , · , < ) is called an Archimedean held. Since the order is total, (Q, · αχχ — — αο =>• x — —α^αο· However, there are equations of the second degree that are unsolvable in some fields. For example, so are the equations — 2 + x 2 = 0 and 1 + x 2 = 0 in . For the first equation, this was proved in part 9 of this subsection. The unsolvability of the second one follows from the rule of signs: since the order is consistent with the operations, if χ £ and y € such that either χ > 0 and y > 0 or χ < 0 and y < 0, then xy > 0. Therefore, x2 > 0 for χφΟ. Remark. In what follows, we consider mostly nonzero groups, rings, and fields.

2.1.4.

Lattices

Ordered sets with binary bounds are naturally associated with algebraic structures. The general theory of such sets is presented in Birkhoff (1979). 1. An upper semilattice (L, V,