Aug 23 Mathematical style, basic logic. Starting Chapter 1.
Aug 25 Distance and limit in the complex plane, 1.3 and following. Homework 1 due Sep 1 Solution
Aug 30 Lim sup
Sep 1 Series, up to 1.6.3 Homework 2 due Sep 8. Solution Read all up to the end of 1.6
Sep 6 1.8 - grouping test, 2.1 - review of complex sequences and series
Sep 8 2.1: Double sums, rearrangement of series. Homework 3 due Sep 15. Solution
Sep 13 Continuing in Ch. 2
Sep 15 Rearrangements, matrices and vectors, Homework 4 due Sep 22 Solution
Sep 18 Total derivative.
Sep 22 Solution to hw4, chain rule. Homework 5 due Sep 29 Solution
Sep 27 Mean value theorem in n dimensions, mixed drivatives.
Sep 29 Taylor theorem in n dimensions. Homework 6 due Oct 6 Solution
Oct 4 Local minima and maxima in n dimensions
Oct 6 Inverse function theorem Homework 7 due Oct 13 Solution
Oct 11 Implicit function theorem
Oct 13 Metric spaces, definition, open sets, examples
Oct 18 Review for midterm Review problems
Oct 20 Midterm. Solution
Oct 25 Metric spaces, limit points, closed sets
Oct 27 Continuous functions on metric spaces. Homework 8 due Nov 3 Solution
Nov 1 Distance from a set, bounded sets, sequences and convergence, uniform limit of continuous functions
Nov 3 Continuing on metric spaces.
Nov 8 Continuing on metric spaces.
Nov 10 Separable spaces. Homework 9 due Nov 17 Solution
Nov 15 Compact spaces.
Nov 17 Compact spaces. Homework 10 due Dec 1 Solution
Nov 29 Compact spaces.
Dec 1 Complete spaces. FCQs. No further homework.
Dec 6 Completion, connected sets, Weierstass theorem, Abel’s theorem (skipped proofs). Application:
P∞ (−1)n+1
= ln 2.
n=1
n
Dec 8 Banach contraction theorem, Arzela-Ascoli theorem. Review for the final
Dec 15 Thursday 3:30-5:30 PL M202 Final - starting from metric spaces final solution
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Analysis prelim syllabus coverage
If you are preparing for the analysis prelim, be sure to study also Rudin’s Principles of Real Analysis
and study also all exercises listed in the syllabus, which often extend the material. Here is where the prelim
topics are expected to be covered, to help with planning. You will need to review ALL in boldface on your
own - do not rely on this class. Be sure to consult the official prelim syllabus at http://www.ucdenver.edu/
academics/colleges/CLAS/Departments/math/program_info/phd/prelims/Pages/AppliedAnalysisSyllabus.
aspx
• Real numbers, infimum, supremum: Rudin 1.1-1.20, 1.23-1.38, exercises Rudin Ch. 1 1-5, 8-19. Construction of reals, infimum, supremum, complex numbers, Euclidean spaces - covered in the prerequisites
• Real Line and Metric Space Topology: Rudin 2.1-2.42, 2.45-2.47. exercises Rudin Ch. 2 1-16, 19-27,
29. Finite, countable, uncountable sets - prerequisites. Metric spaces, open, closed sets, limit points,
compact sets, connected sets, dense sets, separable spaces.
• Numerical Sequences and Series: Rudin 3.1-3.55, Buck 5.2, exercises Rudin Ch. 3 1-18, 20, 21, 2325. Sequences in metric spaces, Cauchy sequences, complete metric spaces, completion. Sequences
and series of reals, harmonic series, convergence tests, power series, Abel’s test (in prerequisites, some
reviewed here). Lim sup and lim inf, rearrangements, multiplication of series.
• Continuous Functions: Rudin 4.1-4.19, 4.25-4.33, Rudin Ch. 4 exercises 1-6, 8-18, 20-25. ε-δ definition
of limit and continuity in metric spaces, continuity and compactness, uniform continuity. Classification
of discontinuities, monotonic functions, limits at infinity (prerequisites).
• Differentiation: Rudin 5.1-5.19, Rudin Ch. 5 exercises 1-7, 9-14, 19, 20, 22-24. Derivative, mean value
theorem, Taylor’s theorem, L’Hospital’s rule - prerequisites.
• Riemann Integration: Rudin 6.1-6.9, 6.12-6.18, 6.20-6.27, Rudin Ch. 6 exercises 1-5. Definition and
properties of Riemann integral - prerequisites.
• Sequences and Series of Functions: Rudin 7.1-7.26. Rudin Ch. 7 exercises 1-13,15-20,24. Uniform
convergence in metric spaces, space C [a, b], uniform convergence and differentiation and integration, Arzela-Ascoli theorem, Weierstrass theorem.
• Power Series: Rudin 8.1-8.5, Rudin Ch. 8 exercises 1-5,7-9. Radius of convergence, uniform convergence, differentiation and integration, Abel’s theorem, multiplication of power series.
• Functions of Several Variables: Rudin 9.1-9.29, 9.39-9.42 Rudin Ch. 9 exercises 1-8,9 (for convex set),
11-16,20,27,30,31. Continuity, partial derivatives, mean value theorem, Taylor’s theorem, inverse and
implicit function theorems.
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