MA REAL ANALYSIS
PURDUE UNIVERSITY SPRING
Instructional Format: Lecture meets times per week for minutes per meeting for weeks.
Credits: .
Text: W. Rudin, Principles of Mathematical Analysis,Third Edition, McGraw-Hill, New York,
Description: Completeness of the real number system, basic topological properties, compactness,
sequences and series, absolute convergence of series, rearrangement of series, properties of contin-
uous functions, the Riemann-Stieltjes integral, sequences and series of functions, uniform conver-
gence, the Stone-Weierstrass theorem, equicontinuity, and the Arzela-Ascoli theorem.
Prerequisites: Upper division undergraduate level course work in Mathematics, General or Up-
per division undergraduate level course work in Engineering, General; for a total of two courses.
Authorized equivalent courses or consent of instructor may be used in satisfying course pre- and
co-requisites.
Topics
○Chapter . The Real and Complex Num-
ber System
–Real number system - (Emphasize
inf, sup)
–Extended real number system
–Euclidean spaces
○Chapter . Basic Topology
–Finite, countable and uncountable
sets
–Metric spaces (Only a few special ex-
amples)
–Compact sets
○Chapter . Numerical Sequences and Se-
ries
–Convergent sequences
–Subsequences
–Cauchy sequences
–lim sup xnand lim inf xn
–Series
–Series with many terms (compari-
son test)
–Absolute and conditional conver-
gence
–Rearrangements
○Chapter . Continuity
–Limits of functions
–Continuous functions
–Continuity and compactness
–Intermediate Value Theorem
○Chapter . The Riemann-Stieltjes Integral
–Definition and existence
–Properties
–Integration and differentiation
○Chapter . Sequences and Series of Func-
tions
–Uniform convergence
–Uniform convergence and continu-
ity
–Uniform convergence and integra-
tion
–Uniform convergence and differen-
tiation
–Equicontinuous families of func-
tions
–Stone-Weierstrass Theorem
○Optional Topics.
–Sets of Lebesgue measure zero
–Characterization of Riemann inte-
grable functions bounded and con-
tinuous a.e.
–Differentiability a.e. of monotone
functions
