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Lecture Notes M 517 Introduction to Analysis Donald J. Estep
Lecture Notes M 517 Introduction to Analysis Donald J. Estep Department of Mathematics Colorado State University Fort Collins, CO 80523 [email protected] http://www.math.colostate.edu/∼estep Contents Chapter 1. Introduction 1 1.1. Review of Real Numbers 1 1.2. Functions and Sets 3 1.3. Unions and Intersection 5 1.4. Cardinality 5 Chapter 2. Metric Spaces 10 2.1. Quick Reivew of Rn 10 2.2. Definition and Initial Examples of a Metric Space 11 2.3. Components of a Metric Space 15 Chapter 3. Compactness 20 3.1. Compactness Arguments and Uniform Boundedness 20 3.2. Sequential Compactness 24 3.3. Separability 25 3.4. Notions of Compactness 28 3.5. Some Properties of Compactness 32 3.6. Compact Sets in Rn 33 Chapter 4. Cauchy Sequences in Metric Spaces 35 4.1. A Few Facts and Boundedness 35 4.2. Cauchy Sequences 36 4.3. Completeness 37 4.4. Cauchy Sequences and Compactness 40 Chapter 5. Sequences in Rn 42 5.1. Arithmetic Properties and Convergence 42 5.2. Sequences in R and Order. 45 5.3. Series 46 Chapter 6. Continuous Funtions on Metric Spaces 48 6.1. Limit of a Function 49 6.2. Continuous Functions 50 6.3. Uniform Continuity 52 6.4. Continuity and Compactness 53 6.5. Rn-valued Continuous Functions 54 Chapter 7. Sequences of Functions and C([a, b]) 57 7.1. Convergent Sequences of Functions 57 7.2. Uniform Convergence: C([a, b]) is Closed, and Complete 61 7.3. C([a, b]) is Separable 63 ii CONTENTS iii 7.4. -
Generalized Tikhonov Regularization: Topological Aspects and Necessary Conditions
Generalized Tikhonov Regularization: Topological Aspects and Necessary Conditions Von der Carl-Friedrich-Gauß-Fakult¨at der Technischen Universit¨atCarolo-Wilhelmina zu Braunschweig zur Erlangung des Grades einer Doktorin der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation von Nadja Sofia Worliczek geboren am 17.08.1978 in Wien/Osterreich¨ Eingereicht am: 17.12.2013 Disputation am: 31.03.2014 1. Referent: Prof. Dr. Dirk Lorenz 2. Referent: Prof. Dr. Thorsten Hohage 2014 ii Contents Contents iii Introductionv 1 Similarity and convergence issues1 1.1 Preliminaries on topological spaces...............1 1.2 Preliminaries on sequential convergence spaces.........5 1.3 Prametrics............................. 11 1.3.1 The prametric topology................. 12 1.3.2 Sequential convergence structures induced by prametrics 17 2 Ill-posed inverse problems and regularization 23 2.1 Inverse problems and terms of solution............. 23 2.2 Well-posedness and stability................... 29 2.3 Regularization........................... 33 3 Generalized Tikhonov regularization 39 3.1 Basic notation........................... 39 3.2 Formalization of Tikhonov setups................ 40 3.3 Some necessary conditions on regularizing Tikhonov setups.. 43 4 Necessary conditions on a common set of sufficient condi- tions 47 4.1 A common set of sufficient conditions.............. 47 4.2 CONV and CONT........................ 50 4.2.1 Bottom slice topologies.................. 51 4.2.2 Necessary conditions................... 55 5 The special case of Bregman discrepancies 59 5.1 Bregman distances - definition and basic properties...... 60 5.2 Bregman prametrics, topological issues and Tikhonov regular- ization............................... 66 iii CONTENTS Conclusion 73 Notation 75 Index 78 Bibliography 81 Zusammenfassung (Summary in German) 87 iv Introduction Context, motivation and subject An inverse problem is the problem of retrieving some quantity of interest from some measured data, that is in some way indirectly related to our quantity of interest, e.g. -
Real Analysis
Real Analysis Andrew Kobin Fall 2012 Contents Contents Contents 1 Introduction 1 1.1 The Natural Numbers . .1 1.2 The Rational Numbers . .1 1.3 The Real Numbers . .6 1.4 A Note About Infinity . .8 2 Sequences and Series 9 2.1 Sequences . .9 2.2 Basic Limit Theorems . 12 2.3 Monotone Sequences . 16 2.4 Subsequences . 19 2.5 lim inf and lim sup . 22 2.6 Series . 23 2.7 The Integral Test . 28 2.8 Alternating Series . 30 3 Functions 32 3.1 Continuous Functions . 32 3.2 Properties of Continuous Functions . 35 3.3 Uniform Continuity . 37 3.4 Limits of Functions . 41 3.5 Power Series . 44 3.6 Uniform Convergence . 46 3.7 Applications of Uniform Convergence . 49 4 Calculus 52 4.1 Differentiation and Integration of Power Series . 52 4.2 The Derivative . 54 4.3 The Mean Value Theorem . 57 4.4 Taylor's Theorem . 61 4.5 The Integral . 63 4.6 Properties of Integrals . 67 i 1 Introduction 1 Introduction These notes were compiled from a semester of lectures at Wake Forest University by Dr. Sarah Raynor. The primary text for the course is Ross's Elementary Analysis: The Theory of Calculus. In the introduction, we develop an axiomatic presentation for the real numbers. Subsequent chapters explore sequences, continuity, functions and finally a rigorous study of single-variable calculus. 1.1 The Natural Numbers The following notation is standard. N = f1; 2; 3; 4;:::g the natural numbers Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g the integers n p o Q = q j p; q 2 Z; q 6= 0 the rational numbers R ? the real numbers Note that N ⊂ Z ⊂ Q ⊂ R. -
Mathematical Analysis Master of Science in Electrical Engineering
Mathematical Analysis Master of Science in Electrical Engineering Erivelton Geraldo Nepomuceno Department of Electrical Engineering Federal University of São João del-Rei August 2015 Prof. Erivelton (UFSJ) Mathematical Analysis August 2015 1 / 89 Teaching Plan Content 1 The Real and Complex Numbers Systems 2 Basic Topology 3 Numerical Sequences and Series 4 Continuity 5 Dierentiation 6 Sequences and Series of Functions 7 IEEE Standard for Floating-Point Arithmetic 8 Interval Analysis Prof. Erivelton (UFSJ) Mathematical Analysis August 2015 2 / 89 References Rudin, W. (1976), Principles of mathematical analysis, McGraw-Hill New York. Lima, E. L. (2014). Análise Real - Volume 1 - Funções de Uma Variável. 12 ed. Rio de Janeiro: IMPA. Overton, M. L. (2001), Numerical Computing with IEEE oating point arithmetic, SIAM. Institute of Electrical and Electronic Engineering (2008), 754-2008 IEEE standard for oating-point arithmetic. Goldberg, D. (1991), What Every Computer Scientist Should Know About Floating-point Arithmetic, Computing Surveys 23(1), 548. Moore, R. E. (1979), Methods and Applications of Interval Analysis, Philadelphia: SIAM. Nepomuceno, E. G (2014). Convergence of recursive functions on computers. The Journal of Engineering q, Institution of Engineering and Technology, 1-3. Prof. Erivelton (UFSJ) Mathematical Analysis August 2015 3 / 89 Assessment Item Value Date Observation N1 - Exam 1 100 02/09/2015 Chapters 1 and 2 N2 - Exam 2 100 14/10/2015 Chapters 3 and 4. N3 - Exam 3 100 04/11/2015 Chapters 5 and 6. N4 - Exam 4 100 02/12/2015 Chapters 7 and 8. Ns - Seminar 100 09/12/2015 Paper + Presentation. Ne - Especial 100 16/12/2015 Especial Exam Table 1: Assesment Schedule 2(N + N + N + N + N ) Score: S = 1 2 3 4 s 500 S + N With N the nal score is: S = e , otherwise S = S: e f 2 f If Sf ≥ 6:0 then Succeed.