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The ETSU Analysis Program: Graduate and Undergraduate; Real , Complex, and Functional A presentation inspired by the 50th anniversary of the publication of Halsey Royden’s Real Analysis. Robert “Dr. Bob” Gardner, Fall 2013 1963 Nationally 1963 Regionally Governor Frank Clement signed the bill renaming “ETSC” “ETSU” on March 5. The name change took effect July 1. Royden’s Real Analysis is published by MacMillan and Company. 1963 1968 1988 2010 Halsey Lawrence Royden (1928-1993) Graduated from Stanford University in 1948 with a bachelor's degree and 1949 with a master's degree. He did his Ph.D. work at Harvard, where he received his doctorate in 1951. He then returned to Stanford as a faculty member. He served as chair of the math department and the departmental historian. He was associate dean for the School of Humanities and Science from 1962 to 1965 and acting dean for 1968-69. He was dean from 1973 to 1981. He was active on the Faculty Senate and was an editor for the Pacific Journal of Mathematics. Halsey Lawrence Royden (cont.) His area of specialty was complex analysis and his Ph.D. advisor was Lars Ahlfors. MathSciNet lists 59 publications for Dr. Royden. Based on class notes he wrote for his class “Theory of Functions of a Real Variable” in the 1950s, he developed his book Real Analysis. He was working on a linear algebra book at the time of his death. Patrick Michael Fitzpatrick Patrick M. Fitzpatrick has a B.A. (1966) and Ph.D. (1971) from Rutgers University. He has been with the University of Maryland since 1975 and was chair of the Department of Math from 1996 to 2007. He is also author of Advanced Calculus, American Mathematical Society (2009). His webpage at the University of Maryland is at: http://www2.math.umd.edu/~pmf/ Real Analysis, Halsey L. Royden (1963) Preface: “This book is the outgrowth of a course at Stanford entitled Theory of Functions of a Real Variable which I have given from time to time during the last ten years. It was designated for first-year graduate students in mathematics and statistics.” Prologue to the Student: “This book covers a portion of the material that every graduate student in mathematics must know.” The ETSU Sherrod Library has a first edition, first printing of Royden’s Real Analysis. It was published by Macmillan and Company (New York) in 1963. The call number is 517.5 R813 (Dewey decimal) or QA331.5.R6 (L.O.C.). The book is stamped “East Tennessee State University” and “ETSU” several places. According to the “Date Due” card in the back, it was first checked out November it was first due on November 19, 1964. It was checked out regularly from then until around 1975, several times in 1981, 1985, 1993, and 1994. The library also has a 1968 printing of the second edition. First Edition (1963) Table of Contents (284 pages) 1. Set Theory Part One. Theory of Functions of a Real Variable 2. The Real Number System 3. Lebesgue Measure 4. The Lebesgue Integral 5. Differentiation and Integration 6. The Classical Banach Spaces Part Two. Abstract Spaces 7. Metric Spaces 8. Topological Spaces 9. Compact Spaces 10. Banach Spaces Part Three. General Measure and Integration Theory 11. Measure and Integration 12. Measure and Outer Measure 13. The Daniell Integral 14. Mappings of Measure Spaces Second Edition (1968) Table of Contents (349 pages) Introductory 1. Set Theory Part One. Theory of Functions of a Real Variable 2. The Real Number System 3. Lebesgue Measure 4. The Lebesgue Integral 5. Differentiation and Integration 6. The Classical Banach Spaces Part Two. Abstract Spaces 7. Metric Spaces 8. Topological Spaces 9. Compact Spaces 10. Banach Spaces Part Three. General Measure and Integration Theory 11. Measure and Integration 12. Measure and Outer Measure 13. The Daniell Integral 14. Measure and Topology 15. Mappings of Measure Spaces Real Analysis 1, 2 and 3 (MHC 620, 621, 622) Academic 1987-88 at Auburn University Taught by Dr. Johnny Henderson Introductory 1. Set Theory Part One. Theory of Functions of a Real Variable 2. The Real Number System 3. Lebesgue Measure 4. The Lebesgue Integral 5. Differentiation and Integration 6. The Classical Banach Spaces Part Two. Abstract Spaces 7. Metric Spaces 8. Topological Spaces 9. Compact Spaces 10. Banach Spaces Part Three. General Measure and Integration Theory 11. Measure and Integration 12. Measure and Outer Measure 13. The Daniell Integral 14. Measure and Topology 15. Mappings of Measure Spaces Linda Lee, Dr. Johnny Henderson, Bob Gardner, circa 1990. Third Edition (1988) Table of Contents (444 pages) Introductory 1. Set Theory Part One. Theory of Functions of a Real Variable 2. The Real Number System 3. Lebesgue Measure 4. The Lebesgue Integral 5. Differentiation and Integration 6. The Classical Banach Spaces Part Two. Abstract Spaces 7. Metric Spaces 8. Topological Spaces 9. Compact and Locally Compact Spaces 10. Banach Spaces Part Three. General Measure and Integration Theory 11. Measure and Integration 12. Measure and Outer Measure 13. Measure and Topology 14. Invariant Measures 15. Mappings of Measure Spaces 16. The Daniell Integral Fourth Edition (2010) Table of Contents (544 pages) Part One Part One. Lebesgue Integration for Functions of a Real Variable 0. Preliminaries on Sets, Mappings, and Relations 1. The Real Numbers: Sets, Sequences, and Functions 2. Lebesgue Measure 3. Lebesgue Measurable Functions 4. The Lebesgue Integral 5. Lebesgue Integration: Further Topics 6. Differentiation and Integration 7. The Lp Spaces: Completeness and Approximation 8. The Lp Spaces: Duality and Weak Convergence Our fall 2012 Real Analysis 1 (MATH 5210) class covered Section 1.4, Chapters 2 and 3, and Sections 4.2, 4.3, and 4.4. Our spring 2013 Real Analysis 2 (MATH 5220) class covered the remainder of Chapter 4, and Chapters 5, 7 and (lightly) 8. Fourth Edition (2010) Table of Contents (544 pages) Part Two Part Two. Abstract Spaces: Metric Spaces, Topological Spaces, Banach Spaces, and Hilbert Spaces 9. Metric Spaces: General Properties 10. Metric Spaces: Three Fundamental Theorems 11. Topological Spaces: General Properties 12. Topological Spaces: Three Fundamental Theorems 13. Continuous Linear Operators Between Banach Spaces 14. Duality for Normed Linear Spaces 15. Compactness Regained: The Weak Topology 16. Continuous Linear Operators on Hilbert Spaces Fourth Edition (2010) Table of Contents (544 pages) Part Three Part Three. Measure and Integration: General Theory 17. General Measure Spaces: Their Properties and Construction 18. Integration Over General Measure Spaces 19. General Lp Spaces: Completeness, Duality, and Weak Convergence 20. The Construction of Particular Measures 21. Measure and Topology 22. Invariant Measures Our Spring 2013 Real Analysis 2 (MATH 5220) class covered Chapters 17, 18, and (lightly) 20. Real Analysis 2 (MATH 5220) Spring 2013 Convergence Theorems!!! The Caratheodory Splitting Condition The Dual Space of Lp is Lq. Riesz Representation Theorem History of the ETSU Math Graduate Program The Beginning our Graduate Program The 1951-52 East Tennessee State College Bulletin introduced the “Graduate Division” and reports the following (pages 140-141): The program involves a Master of Arts in education. “Minor Fields” listed by name are English, social studies, biology, and chemistry. The First “Math Thesis” According to the ETSU Sherrod Library online catalog (http://libraries.etsu.edu/; search in the “ETSU Sherrod Library-Theses & Diss.” pulldown menu) and the Listing of Master’s Theses, East Tennessee State University, 1951-1976 (which lists 89 departmental theses), the first thesis to fall under the heading of math was: Florence Bogart, “Making First-Year Algebra More Meaningful” (1951). The 1952-53 East Tennessee State College Bulletin lists math as a “Minor Field” in the Master or Arts in education degree and gives the first post-calculus analysis-type class (as well as the first appearance of a modern algebra-type class): 502. Infinite Series. (3 hours) A Study will be made of theory and application of infinite series, convergence, divergence, ratio and comparison tests power series, expansion of functions in series. Prerequisite: Math. 303 [Calculus 3] 503. Introduction to Modern Algebra. (3 hours) This is a study of polynomials and fundamental properties, linear dependents, matrices, invariants, bilinear forms, and selected topics from the theory of numbers and finite groups. Prerequisite: Math. 303. The faculty listed for the Math Department are: “Mr. Carson, Miss Cloyd, Mr. Jasper, Miss Smith, and Mr. Stollard.” The 1953-54 East Tennessee State College Bulletin has the addition of Advanced Calculus: 511S, 512S. Advanced Calculus . (3 hours) Supplementary study of calculus, including partial derivatives with applications to surfaces and space curves, Taylor’s Theorem and relative extremes of functions of several variables, line and surface integrals, improper integrals, indeterminate forms. The 1955-56 East Tennessee State College Bulletin has the addition of: 551, 552. Functions of a Complex Variable . (3 hours) 595. Methods of Research. The 1956-57 East Tennessee State College Bulletin has the addition of: 521S. Elementary Math from an Advanced Standpoint . (3 hours) An ‘S’ means that the class can be taken by qualified seniors. The 1960-61 East Tennessee State College Bulletin includes: http://www.universetoda y.com/102783/kapowkeck-confirms-puzzlingelement-of-big-bangtheory/ 412G-413G. Introduction to Modern Algebra. (3 hours each quarter) Construction of the number systems in algebra; groups, rings, and fields; polynomials. 541-542-543. Modern Algebra. (3 hours each quarter) Prerequisite: Math 412G [Introduction to Modern Algebra 1]. Theory of groups, rings, integral domains, and fields; polynomials; vector spaces, and theory of ideals. An ‘G’ denotes an undergraduate class which graduates can take for credit. The 1960-61 East Tennessee State College Bulletin (continued – Real and Complex): 521-522-523. Foundations of Analysis. (3 hours each quarter) Prerequisite: Math. 422G [Advanced Calculus 2]. A rigorous study of the real and complex number systems, elements of set theory, numerical sequences and series, continuity, differentiation, the Riemann-Stieltjes Integral, sequences and series of functions, further topics in the theory of series, functions of several variables, and the Lebesque Theory. 551-552-553. Functions of a Complex Variable. (3 hours each quarter) Prerequisite: Math 423G [Advanced Calculus 3]. Complex numbers, analytic and elementary functions, integrals, power series, residues and poles, conformal mapping and applications. 555-556-557. Functions of a Real Variables. (3 hours each quarter) Prerequisite: Math. 423 G [Advanced Calculus 3]. The real number system, theory of point sets, rigorous investigation of many questions arising in calculus, Lebesque integrals and infinite series. The 1960-61 East Tennessee State College Bulletin (continued – Topology and Research): 411G. Introduction to Topology . (3 hours) A basic course in the properties of a space which are invariant under continuous transformations. Set topology, homology, homotopy, fixed point theorems, and manifolds. 575-576-577. Topology. (3 hours each quarter) Prerequisite: Math. 411G [Intro to Topology] and Math. 523 [Foundations of Analysis 2]. An Introduction to the study of geometric properties that depend only on continuous structure and not on size or shape. Topics will be selected from the following: point sets, cardinal numbers, neighborhood spaces, continuous mappings, connectivity, network, polyhedra. 595. Introduction to Mathematical Research. (1 hour) 596. The Thesis. (4-6 hours) Modern Algebra at ETSU The Evolution of our Modern Algebra Sequences 412G-413G. Introduction to Modern Algebra Change ‘5’ to ‘4/5’ and add a ‘7’ 4127/5127, 4137/5137. Introduction to Modern Algebra 1 and 2. 541-542-543. Modern Algebra. Add a ‘0’, drop third class 5410, 5420. Modern Algebra 1 and 2. Analysis at ETSU The Evolution of our Analysis Sequences 521-522-523. Foundations of Analysis. Change ‘5’ to ‘4/5’ and add a ‘7’ 4217/5217, 4227/5227. Analysis 1 and 2. 555-556-557. Functions of a Real Variable. Add a ‘0’, drop third class 5210, 5220. Real Analysis 1 and 2. The Evolution of our Complex Analysis Classes 551-552-553. Functions of a Complex Variable. ? 4337/5337. Complex Variables. Add a ‘0’, drop third class 5510, 5520. Complex Analysis 1 and 2. The Evolution of our Topology Class 411G. Introduction to Topology. 575-576-577. Topology. 5350. Topology. 4357/5357. Introduction to Topology. Searching the ETSU Sherrod Library online catalog (http://libraries.etsu.edu/; search in the “ETSU Sherrod Library-Theses & Diss.” pulldown menu) for departmental theses (search for “East Tennessee State University Dept of Mathematics”) reveals a total of 253 theses (as of September 29, 2013). By my classification, the following 8 fall in the area analysis: 1. Applications of Lebesgue Theory in Probability and Statistics, David Burt, 1994. 2. Bounds on the Location of Zeros of Polynomials with Complex Coefficients, Atif Abueida, 1996. 3. Construction of Piecewise Linear Wavelets, Jiansheng Cao, 2002. 4. Construction of Prewavelets over Triangulations, Yuchun Mu, 1999. 5. Development of the Real Number System, Garda Browning, 1970. 6. Extensions of the Cayley-Hamilton Theorem with Applications to Elliptic Operators and Frames, Alberto Mokak Teguia, 2005. 7. On the Construction of Linear Prewavelets over a Regular Triangulation, Qingbo Xue, 2002. 8. Lp Inequalities and Rate of Growth Inequalities for a Polynomial and its Derivative, Amy Weems, 1996. 9. Strictly Topological Properties, Kai Dai, 1966. Undergraduate/Graduate Analysis Classes • Analysis 1 (MATH 4217/5217): In a setting of the real numbers, a study of sets, functions on sets, algebra and order properties of the real numbers, sequences, sequences of functions, continuous functions, limits of functions, function approximation.* • Analysis 2 (MATH 4227/5227): A study of the derivative and integral of functions of one variable, the Riemann- Stieltjes integral, series, series of functions, Fourier series* • Complex Variables (MATH 4337/5337): Complex numbers and their algebra, complex differentiation and integration, analytic and elementary functions, residues and power series. • Introduction to Topology (MATH 4357/5357): Open and closed sets, continuous functions, metric spaces, connectedness, continuous functions, metric spaces, connectedness, compactness, the real line, and the fundamental group. Descriptions from the 2013-14 Graduate Catalog. *The description of Analysis 1 and Analysis 2 are different in the Undergraduate Catalog. Graduate Analysis Classes • Real Analysis 1 (MATH 5210): The real number system, sequences, functions, Lebesgue measure, measurable functions, the Lebesgue integral, differentiation. • Real Analysis 2 (MATH 5220): Classical Banach spaces, metric spaces, topological spaces, Riesz Representation theory, Hilbert spaces, general measure and integration. • Complex Analysis 1 (MATH 5510): Complex numbers, continuity and derivatives of functions, power series, Laurent series, transcendental functions, line and contour integrals, Cauchy integral formula, Maximum Principle, applications. • Complex Analysis 2 (MATH 5520): Residues and integration, harmonic functions, conformal mapping and applications, Laplace transform, Riemann surfaces. • Introduction to Functional Analysis (MATH 5740): An introduction to the standard topics of functional analysis are given. Properties of normed linear spaces, Banach spaces, and Hilbert spaces are studied. The Hahn-Banach Theorem is addressed. Spectral theory and compact operators are introduced. A knowledge of measure theory is not assumed and any needed measure theoretic results are presented in the course.* Descriptions from the 2013-14 Graduate Catalog. *Proposed description for this experimental course. Analysis 1 and 2 (MATH 4217/5217 and 4227/5217) An Introduction to Analysis, 2nd Edition 1. The Real Number System. 2. Sequences of Real Numbers. 3. Topology of the Real Numbers. 4. Continuous Functions. 5. Differentiation. 6. Integration. 7. Series of Real Numbers. 8. Sequences and Series of Functions. 9. Fourier Series. Taught primarily by Gardner 1993-2007, but also taught by J. Knisley, Norwood, Godbole, and, starting in fall 2013, Helfgott. Other texts have been used. Complex Variables (MATH 4337/5337) Complex Variable and Applications, 8th Edition 1. Complex Numbers. 2. Analytic Functions. 3. Elementary Functions. 4. Integrals. 5. Series. 6. Residues and Poles. 7. Applications of Residues. 8. Mapping by Elementary Functions. 9. Conformal Mapping. 10. Applications of Conformal Mapping. Taught primarily by J. Knisley over the past 20 years, but also by Gardner and Norwood. Real Analysis 1 and 2 (MATH 5210 and 5220) Spring 2002 - Spring 2006 Real Analysis with an Introduction to Wavelets and Applications 1. Fundamentals. 2. The Theory of Measure. 3. The Lebesgue Integral. 4. Special Topics in Integral and Application. 5. Vector Spaces, Hilbert Spaces, and the L2 Space. Taught as follows: • Spring 2002 (Gardner, Part 1). • Fall 2002, spring 2003 (Hong, Parts 1 & 2). • Spring 2004 (Hong, Part 1). • Fall 2004, Spring 2005 (Hong, Parts 1 & 2). • Spring 2006 (Gardner, Part 1). Complex Analysis 1 and 2 (MATH 5510 and 5520) Functions of One Complex Variable, 2nd Edition I. The Complex Number System. II. Metric Spaces and the Topology of C. III. Elementary Properties and Examples of Analytic Functions. IV. Complex Integration. V. Singularities. VI. The Maximum Modulus Theorem VII. Compactness and Convergence in the Spaces of Analytic Functions. IX. Analytic Continuation and Riemann Surfaces. Introduction to Functional Analysis (MATH 5740) A First Course in Functional Analysis 1. Linear Spaces of Operators. 2. Normed Linear Spaces: The Basics. 3. Major Banach Space Theorems. 4. Hilbert Spaces. 5. Hahn-Banach Theorem. 6. Duality. 8. The Spectrum. 9. Compact Operators. Dr. Bob’s Analysis Award Dr. Bob’s Sporadic Analysis Award An Unofficial, Unrecognized, Unprestigious Award The first "Dr. Bob's Analysis Award" was granted to Ed Snyder on April 30, 2013. Dr. Bob’s Sporadic Analysis Award New Categories (A Subversive Recruiting Tool) Standard Version: The standard version of the award is given to those taking both the Real Analysis and Complex Analysis sequences with Dr. Bob. Gold Version: The gold version of the award is given to those taking both the Real Analysis and Complex Analysis sequences with Dr. Bob, along with the newly created "Introduction to Functional Analysis." Platinum Version: The platinum version of the award (described by some as the holy grail of unprestigious, unrecognized analysis awards) is given to those taking the Real Analysis sequence, the Complex Analysis sequence, and Introduction to Functional Analysis with Dr. Bob, along with writing an analysis-based thesis with Dr. Bob. Keep taking those analysis classes! Stephan Banach 1892-1945 Some References 1. An online obituary of Halsey Royden: http://news.stanford.edu/pr/93/930824Arc3164.html 2. An Introduction to Analysis, 2nd edition, by J. R. Kirkwood, PWS Publishing Company and Waveland Press (1995). 3. Real Analysis, Fourth Edition, by H.L. Royden and P.M. Fitzpatrick, Prentice Hall (2010). 4. A First Course in Functional Analysis, by S. David Promislow, John Wiley & Sons Publications (2008). 5. Real Analysis with an Introduction to Wavelets and Applications, by D. Hong, J. Wang, and R. Gardner, Academic Press/Elsevier Press (2005). 6. Functions of One Complex Variable, Second Edition, by John Conway, Springer-Verlag (1978). 7. Complex Variables and Applications, 8th edition, by James Brown and Ruel Churchill, McGraw-Hill (2008).