MATH 4210 Foundations of Analysis
Spring 2018
Lectures 2:30-3:20 pm MWF in 113 MLH
Discussion 2:00-2:50 pm Th in 213 MLH

Instructor:  Oguz Durumeric

Office location: B20F MLH

Office hours:  Mon, Wed, Thu 9:00-10:20

Phone: (319) 335-0774

Email: oguz-durumeric@uiowa.edu

Website: http://www.math.uiowa.edu/~odurumer/

TA: Chistopher Adams (Meeting times 2:00-2:50 Th, in 213 MLH)

DEO Contact Information: Professor Maggy Tomova, 14 MLH, maggy-tomova@uiowa.edu

Prerequisites for MATH 4210: MATH 3770 or Graduate standing

Catalog Description of the course: Introduction to fundamental ideas of analysis; emphasis on understanding and constructing definitions, theorems, and proofs; real and complex numbers, set theory in metric spaces, compactness and connectedness, sequences, Cauchy sequences, series, and continuity; elements of differential and integral calculus; sequences and series of functions; modes of convergence; equicontinuity; serves as a bridge between MATH:3770 and MATH:5200.

·        Objectives and Goals of the course: Strong emphasis on formal reasoning, understanding and constructing definitions, theorems, and proofs

·        Real and Complex Numbers:  Development of the real numbers from the rationals.  Extended real numbers.  Complex Numbers.  Euclidean Spaces.

·        Set Theory: Countable/Uncountable Sets.  Metric Spaces.  Open and closed sets in metric spaces.  Compactness with finite subcovers. Weierstrass theorem.  Cantor Set.  Connectedness.

·        Sequences: Convergence, subsequences, and Cauchy sequences in R and in general metric spaces.  Limsup and liminf.  Completeness.  Series, partial sums, convergence and absolute convergence.  Power series and radius of convergence.

·        Continuity:  Continuity, relationship of continuity and compactness, relationship of continuity and connectedness.

·        Calculus: Derivatives.  Mean value theorem.  Taylor's theorem.  Riemann-Stieltjes integral. Fundamental Theorem.  Functions of bounded variation.

·        Sequences and Series of Functions: Modes of convergence: Pointwise, Uniform, Lp.  Relationship of modes of convergence and continuity, integration and differentiation.  Stone-Weierstrass Theorem.  Equicontinuity and Arzela-Ascoli theorem.

·        As much as time permits, Functions of several variables, Implicit and Inverse function Theorems

Textbook: Principles of Mathematical Analysis, by Walter Rudin ISBN 978-0-0415-42356 McGraw Hill.

SYLLABUS

LECTURE NOTES

ASSIGNMENTS

EXAM INFORMATION and SUPPLEMENTARY FILES
Solution to Midterm 2