MATH 447- B13 and C13- Introduction to Real Analysis

Time and Location:  B13: MWF 10:00-10:50am 106B6  Engeneering Hall,   C13: MWF 14:00-14:50 1060 Lincoln Hall

Instructor: Marius Junge

Office hours:  Thursday 5-6

Course description
:

Introduction to real analysis is a real math course: We rigorously combine content and proofs. Your job is to learn writing and unerstanding  proofs in combination with century old matematically concepts developped to make math easier.

In contrast to calculus this course relies on some level of abstract math to clarify the specfics of the real numbers. In parctive this means we start with metric spaces, the notion of distance, open and closed balls. Then we proceed with notions of convergence for individual seguences and sequences of functions. In this light we discover the notion of uniform convergence in connection with compact sets. All this is deveilopped to establisch the existence of integrals. At this point, we could then start to do `real analysis', but then the semester is over anyway, ...

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.

Notes

II: Sequences

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

Notes

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.

Notes

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.

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

Homework   (20%)

Midterm1   September 30 (25%)
Midterm2   November 4  (25%)

Final:  see uncombined schedule (30%)

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

Additional Material: See Box