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

Office location: B20F MLH

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

Email: oguz-durumeric@uiowa.edu

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

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**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.

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

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

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

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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.

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

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

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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.

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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.

LECTURE NOTES

ASSIGNMENTS

EXAM INFORMATION and SUPPLEMENTARY FILES

Solution to Midterm 2