Time: MWF 10:00-10:50pm
Location: Altgeld Hall 243Instructor: Marius Junge
I: Real Numbers
Natural numbers, abelian groups, Grothendiecks construction, integers, fields, rational numbers, ordered fields, completeness, Peano's axiom, uncountability of real numbers.
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.
Rolle's Lemma and Mean value theorem, application differentiation of power series.
Definition, interching limits, fundamental theorem, application to power series.
Practise problems for final and solutionsBooks:
Elementary Analysis: The Theory of Calculus by Kenneth Ross, 2 edition, SpringerGrading:
Midterm1 September 26 (25%)
Midterm2 October 26 (25%)
Final: see uncombined schedule
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