Office location: B20F MLH
Office hours: Mon, Wed, Thu 9:00-10:20
Email: oguz-durumeric@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.
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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.