University of Houston
Department of Mathematics
MATH 4331; 4332, Introduction to Real Analysis
Prerequisites: MATH 3334 or consent of the instructor
Recommended text: “Principles of Mathematical Analysis” , 3rd Ed., by Walter Rudin, McGraw-Hill, 1976
Course Description: This course is our senior level sequence in Real Analysis. The function of this sequence is to give the student a background in the underlying theoretical basis for the Calculus and to prepare them for graduate level coursework in analysis. It is designed to be a theorem-proving course with a particular emphasis on topological notions as they relate to the study of analysis. Metric spaces play a central role in analysis and the basic properties of completeness, compactness, connectedness and their interplay with continuity will be emphasized. The Heine-Borel and Bolzano-Weierstrass Theorems will be proved. Convergence of sequences in metric spaces and convergence of sequences of functions lead up to Ascoli's Theorem and the Stone-Weierstrass Theorem. Riemann-Stieltjes and Lebesgue integration for real-valued functions of a real variable will be developed.
SUGGESTED SYLLABUS
Math 4331
Chapter 2: Basic Topology of Metric Spaces: Should be covered in its entirety, including the Heine-Borel and Bolzano-Weierstrass Theorems.
Chapter 3: Sequences and Series: To include convergent sequences, Cauchy sequences, completeness, lim-sup and lim-inf. It is recommended to cover the completion of a metric space, which appears as a problem in Rudin. Need not spend much time going over tests for convergence of numerical series, but may include power series.
Chapter 4: Continuity: This chapter is short and should be covered in its entirety.
Chapter 6: The Riemann-Stieltjes Integral
Math 4332
Chapter 7: Sequences and Series of Functions: Should cover uniform convergence, Ascoli's Theorem, the Stone-Weierstrass Theorem. A review of Taylor series in Chapter 5 and Power Series in Chapter 8 would be useful.
Chapter 9: Functions of Several Variables: One can do a careful treatment of the Inverse and Implicit Function Theorems together with the Contraction Mapping Principle. The rest of this Chapter is at the instructor's discretion. Note that 9.10-9.21 should have been covered in Math 3334, so, perhaps, one could move through this quickly.
Chapter 11: Lebesgue Theory: The Lebesgue Theory for the real line only.
Chapter 1: It is hoped that a degree in Mathematics would include exposure to the Dedekind approach to the development of the real number system, as time permits.
Policy on Incompletes: See p. 62 of the Undergraduate Catalog.
Whenever possible, and in accordance with 504/ADA guidelines, the University of Houston will attempt to provide reasonable academic accommodations to students who request and require them. Please call 713-743-5400 for more assistance.