I. Number and Title of Course

MAT 611 Topics in Real Analysis

A. This course is intended to provide a beginning graduate course in the theory of functions of a real variable, primary for graduate students who have not completed the equivalent of MAT 417 (Introduction to Real Analysis I) as undergraduates.

B. This course is intended to provide students with opportunities to learn proofs of the theorems of calculus in a formal, rigorous way.

A. To acquaint students with the foundations of the real number system and its

axioms

B. To present formal proofs of the theorems in topics such as limits, functions,

continuity, sequence, series, integration

C. To strengthen the students' skills in constructing proofs

A) The Real Numbers

1. The field, order, and completeness axioms

2. Applications of the least upper bound properties

3. The extended real number system

B) Basic Topology

1. Open and closed sets in R

2. Metric space

3. Euclidean space

4. Compact sets, perfect sets, and connected sets

5. Heine - Borel theorem

C) Sequences

1. Basic limit theorems

2. Monotone sequences and the number e

3. Bolzano - Weierstrass theorem

4. The Cauchy condition

5. Iim sup and km inf

D) Continuous Functions

1. Continuity and uniform continuity

2. Continuity and compactness

3. Continuity and connectedness

E) Differentiation

1. Definition and basic theorems

2. Mean value theorem

3. L'Hospital theorem

4. Taylor theorem

F) The Riemann-Stieltjes Integral

1. Definition and existence of the integral

2. Properties of the integral

3. Integration and differentiation

4. Rectifiable curves

G) Sequence and Series of Functions

1. Point wise convergence and uniform convergence

2. Integration and differentiation of uniform convergent series

3. Applications to power series

4. Abel's limit theorem (optional)

5. Summability method and Tauberian theorem (optional)

H) Some Special Functions (optional)

1. Fourier series

2. The gamma function

I) The Lebesque Theory

1. Set function

2. Construction of the Lebesque measure

3. Measure spaces

4. Measurable functions

5. Simple functions

6. Lebesque integral

7. Comparison with the Riemann Integral

Aliprantis, C. D., Principles of Real Analysis, Harcourt Brace Jovanovich, Boston, 1990

Aliprantis, C. D., Problems in Real Analysis, Academic Press, Boston, 1990

Apostol, T., Mathematical Analysis, Addison Wesley, Reading, MA, 1984

Bartle, R., and Sherbert, D., Introduction to Real Analysis, John Wiley & Sons, Inc., New York, 1992

Boas, R. P., A Primer of Real Functions, Mathematical Association of America, Washington, 1982

Buck, R., Advanced Calculus, McGraw Hill, New York, 1965

Dudley, R. M., Real Analysis and Probability, Wadsworth & Brooks/Cole Advanced Books and Software, Pacific Grove, CA, 1989

Eaves, E. D. and Carruth, J. H., Introductory Mathematical Analysis, Wm. Brown, Dubuque, IA, 1985

Falland, G., Real Analysis, Wiley, New York, 1984

Hauser, N. B., Real Analysis, Dover Publications, Mineola, NY, 1991

Heider, L. and Simpson, J., Theoretical Analysis, W. B. Saunders, Philadelphia, PA, 1967

Johnsonbaugh, R. and Pfaffenberger, W., Foundations of Mathematical Analysis, Dekker, New York, 1981

Kolmogorov, A. and Fomin, S., Introductory Real Analysis, Dover, Mineola, NY, 1970

Lang. S., Analysis l and ll, Addison Wesley, Reading, MA, 1968

Light, W. A., An Introduction to Abstract Analysis, Chapman & Hall, New York, 1 990

Markarov, B. M., et. al., Selected Problems in Real Analysis, American Mathematical Society, Providence, Rl, 1992

Ruckle, W., Modern Analysis, PWS-Kent, Boston, 1991

Rudin, W., Real and Complex Analysis, McGraw Hill, New York, 1987

Rudin, W., Principles of Mathematical Analysis, McGraw New York, 1976

Stanel, D. C., Real Analysis With Point? Set Topology, Dekker, New York, 1987

Stromberg, Karl R., Introduction to Classical Real Analysis, Chapman & Hall, New York, 1981

Torchinsky, A., Real Variables, Addison-Wesley, Reading, MA, 1988

Lectures, discussions, problem sets, student presentations, examinations

Three semesters of an undergraduate calculus sequence

Three semester hours (3:O)

This course proposal was examined in accord with recommended procedures and was approved by the Mathematics Graduate Faculty on May 13,1993.      ____________________________________Chairman

X. Catalogue Description

MAT 611 - Introduction to Real Functions ? The real numbers, basic topology, sequences, continuous functions, differentiation, the Reimann-StieltJe's integral, sequence and series of functions, some special functions, the Lebesque theory.

A masters degree in mathematics with a concentration in analysis is the minimum formal education requirement.

Several members of the mathematics department meet this requirement. Some of the current faculty who could teach this course are:

Barback, J. (Ph.D., Rutgers University, 1969)

Carbonara, J. (Ph.D., University of California at San Diego, 1992)

Chen, J. (Ph.D., University of Rochester, 1969)

Cunningham, D. (Ph.D., University of California at Los Angeles, 1991)

Green, A., (Ph.D., Syracuse University, 1972)

Guyker, J., (Ph.D., Lehigh University, 1970)

Ohm, K., (Ph.D., Syracuse University, 1967)

Slivka, J., (Ph.D. State University of New York at Buffalo, 1969)

Stern, S., (PhD., University at Buffalo, 1962)