MATH 447- C13- Introduction to Real Analysis

Time:  MWF 10:00-10:50pm

Location:  Altgeld Hall  243 

Instructor: Marius Junge    

Grader:  Mary Gramcko Tursi 

Office hours:  Tuesday 5-6

Course description

Introduction to real analysis is a gateway. The idea is to find  balance between rigorous proofs and real understanding - this principle is the core of mathematics at all levels.

Be prepared to learn to write proofs.

Be prepared to accept a little absract but clarifying approach to well known, and not so well known topics related to calculus.

I: Real Numbers 

Natural numbers, abelian groups, Grothendiecks construction, integers, fields, rational numbers, ordered fields, completeness, Peano's axiom, uncountability of real numbers. 


II: Sequences  

Limits, monotone sequences, subsequences, Bolzano-Weierstrass, limsup and liminf, application to continuous functions. 


III: Metric spaces

Metric spaces, Cauchy sequences, completeness, sequential compactness and total boundedness, open, closed  and compact sets, application to Heine Borel and continuity of inverses. Connectes sets, intermediate value theorem. 



Compact metric spaces

Connected metric spaces

Spaces of continuous functions

III:  Spaces of continuous functions

Uniform continuity, C(K) is a complete metric space, Dini's theorem, application: interchanging differentiation and limit.

IV: Differentiation 

Rolle's Lemma and Mean value theorem, application differentiation of power series. 

V: Integration

Definition, interching limits, fundamental theorem, application to power series. 

 Practise problems for final and solutions

Elementary Analysis: The Theory of Calculus by Kenneth Ross, 2 edition, Springer

Homework   (20%)

Midterm1   September 26 (25%)
Midterm2   October  26 (25%)

Final:  see uncombined schedule

Video lecture

Homework: See folder in Box, contact me if not invited

Additional Material: See Box

Practiceproblem 2 solutions

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