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.

19681963

2010

1988

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.

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.

His area of specialty was complex analysis and his Ph.D. advisor was Lars Ahlfors.

Halsey Lawrence Royden (cont.)

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/

Patrick Michael Fitzpatrick

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.

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.”

Real Analysis, Halsey L. Royden (1963)

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 Spaces10. Banach Spaces

Part Three. General Measure and Integration Theory11. Measure and Integration12. Measure and Outer Measure13. The Daniell Integral14. Mappings of Measure Spaces

First Edition (1963)Table of Contents (284 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 Spaces10. Banach Spaces

Part Three. General Measure and Integration Theory11. Measure and Integration12. Measure and Outer Measure13. The Daniell Integral14. Measure and Topology15. Mappings of Measure Spaces

Second Edition (1968)Table of Contents (349 pages)

Real Analysis 1, 2 and 3 (MHC 620, 621, 622)Academic 1987-88 at Auburn University

Taught by Dr. Johnny Henderson

Linda Lee, Dr. Johnny Henderson, Bob Gardner, circa 1990.

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 Spaces10. Banach Spaces

Part Three. General Measure and Integration Theory11. Measure and Integration12. Measure and Outer Measure13. The Daniell Integral14. Measure and Topology15. Mappings of Measure Spaces

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 Spaces10. Banach Spaces

Part Three. General Measure and Integration Theory11. Measure and Integration12. Measure and Outer Measure13. Measure and Topology14. Invariant Measures15. Mappings of Measure Spaces16. The Daniell Integral

Third Edition (1988)Table of Contents (444 pages)

Part One. Lebesgue Integration for Functions of a Real Variable0. Preliminaries on Sets, Mappings, and Relations1. The Real Numbers: Sets, Sequences, and Functions2. Lebesgue Measure3. Lebesgue Measurable Functions4. The Lebesgue Integral5. Lebesgue Integration: Further Topics6. Differentiation and Integration7. The Lp Spaces: Completeness and Approximation8. The Lp Spaces: Duality and Weak Convergence

Fourth Edition (2010)Table of Contents (544 pages)

Part One

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.

Part Two. Abstract Spaces: Metric Spaces, Topological Spaces, Banach Spaces, and Hilbert Spaces 9. Metric Spaces: General Properties10. Metric Spaces: Three Fundamental Theorems11. Topological Spaces: General Properties12. Topological Spaces: Three Fundamental Theorems13. Continuous Linear Operators Between Banach Spaces14. Duality for Normed Linear Spaces15. Compactness Regained: The Weak Topology16. Continuous Linear Operators on Hilbert Spaces

Fourth Edition (2010)Table of Contents (544 pages)

Part Two

Part Three. Measure and Integration: General Theory17. General Measure Spaces: Their Properties and Construction18. Integration Over General Measure Spaces19. General Lp Spaces: Completeness, Duality, and Weak Convergence20. The Construction of Particular Measures21. Measure and Topology22. Invariant Measures

Fourth Edition (2010)Table of Contents (544 pages)

Part Three

Our Spring 2013 Real Analysis 2 (MATH 5220) class covered Chapters 17, 18, and (lightly) 20.

Convergence Theorems!!!

The Dual Space of Lp is Lq.

The Caratheodory Splitting Condition

Riesz Representation Theorem

Real Analysis 2 (MATH 5220)Spring 2013

History of the ETSU Math Graduate

Program

The Beginning our Graduate ProgramThe 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.

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:

The First “Math Thesis”

Florence Bogart, “Making First-Year Algebra More Meaningful” (1951).

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 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):

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.

An ‘S’ means that the class can be taken by qualified seniors.

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)

The 1960-61 East Tennessee State College Bulletin includes:

412G-413G. Introduction to Modern Algebra. (3 hours each quarter)Construction of the number systems in algebra; groups, rings, and fields; polynomials.

An ‘G’ denotes an undergraduate class which graduates can take for credit.

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.

http://www.universetoday.com/102783/kapow-keck-confirms-puzzling-element-of-big-bang-theory/

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 – Real and Complex):

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)

The 1960-61 East Tennessee State College Bulletin (continued – Topology and Research):

Modern Algebra at ETSU

412G-413G. Introduction to Modern Algebra

541-542-543. Modern Algebra.

4127/5127, 4137/5137. Introduction to Modern

Algebra 1 and 2.

Change ‘5’ to ‘4/5’ and add a ‘7’

Add a ‘0’, drop third class

5410, 5420. Modern Algebra 1

and 2.

The Evolution of our Modern Algebra Sequences

Analysis at ETSU

521-522-523. Foundations of

Analysis.

555-556-557. Functions of a Real

Variable.

4217/5217, 4227/5227. Analysis 1 and 2.

Change ‘5’ to ‘4/5’ and add a ‘7’

Add a ‘0’, drop third

class

5210, 5220. Real Analysis 1 and 2.

The Evolution of our Analysis Sequences

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 Complex Analysis Classes

411G. Introduction to Topology.

575-576-577. Topology.

4357/5357. Introduction to Topology.

The Evolution of our Topology Class

5350. Topology.

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.

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:

• 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.

Undergraduate/Graduate Analysis Classes

Descriptions from the 2013-14 Graduate Catalog.*The description of Analysis 1 and Analysis 2 are different in the Undergraduate Catalog.

• 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.*

Graduate Analysis Classes

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.

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).

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.

Real Analysis 1 and 2 (MATH 5210 and 5220)Spring 2002 - Spring 2006

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.

Complex Analysis 1 and 2 (MATH 5510 and 5520)

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 AwardAn 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 AwardNew Categories (A Subversive Recruiting Tool)

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."

Standard Version: The standard version of the award is given to those taking both the Real Analysis and Complex Analysis sequences with Dr. Bob.

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.

Karl Weierstrass

1815-1897

Carl Gauss

1777-1855

Augustin Cauchy1789-1857

Bernhard Riemann1826-1866

Frigyes Riesz

1880-1956

David Hilbert1862-1943

Henri Lebesgue1875-1941

Stephan Banach1892-1945Keep taking those

analysis classes!

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).