Analysis I (Fall 2017)


Instructor: Paul E. Hand

Office: Duncan 3086

Email: hand [at]

Lectures: TR, 10:50-12:05 in DCH 1046

Office Hours: Mondays 4-5:30pm and by appointment


The objectives of this course are: (1) to provide you with a firm understanding of the basic ideas of mathematical analysis; (2) to improve your proof writing skills up to a professional level; (3) to improve problem solving skills needed for the CAAM analysis qualifier exam; and (4) to improve your mathematical communication skills. The specific analysis topics covered include Real numbers, completeness, sequences and convergence, compactness, continuity, the derivative, the Riemann integral, the fundamental theorem of calculus. vector spaces, dimension, linear maps, inner products and norms, derivatives in R^d, Inverse Function Theorem, Implicit Function Theorem, Multiple Integration, Change of Variable Theorem. See the syllabus.

Text Book: Undergraduate Analysis by Lang

Homeworks: You will have about three homework problems a week. They will be due on Tuesdays. Your proofs should be your absolute best quality work. They will each be graded out of one point, taking into account the correctness and the exposition. After you receive your graded solution, you may resubmit it once (at the next homework due date). The grade of the second submission will be final. Two homework assignments will be pledged and will serve as simulated qualifier exams. Here are some examples of the quality of proofs expected. Two homework assignments will be pledged and will serve as simulated qualifier exams.

Class Time: Class will primarily consist of a discussion and Q&A on the topics assigned that week. It will also include some problem solving and presentation of solutions to homework problems. You are expected to read and think deeply about the day’s topics before the beginning of class. During the discussion, I will choose students randomly to present and answer questions on that day’s topics. See these comments for the sorts of things you should think about.

Outside resources: You are not allowed to use the Problems and Solutions book accompanying Lang’s Undergraduate Analysis text for any of the homeworks.

Disabilities: Any student with a disability needing academic accommodations is requested to speak with me as soon as possible. All discussions will remain confidential. Students should also contact Disability Support Services in the Ley Student center.



Event Date Related Documents
HW 1Aug 29 in classProblems.
HW 2Sep 5 in classProblems.
HW 3Sep 12 in classProblems.
HW 4Sep 26 in classProblems.
HW 5Oct 3 in classProblems.
Pledged HW 6Oct 19 in classExam 2017 with solutions. Practice Exam 2016. Practice Exam 2015 with solutions. Practice Exam 2014.
HW 7Oct 24 in classProblems.
HW 8Oct 31 in classProblems
HW 9Nov 7 in classProblems.
HW 10Nov 14 in classProblems.
Pledged HW 11Nov 30 in classExam 2017. Final from 2016. Final from 2014. Final from 2015

Lecture Notes

Topics and dates HAVE BEEN UPDATED AS OF OCT 11, 2017
Day Topics Reference (Lang) Class notes
Aug 22,24Real numbers. Cauchy sequences, Bolzano-Weierstrass, liminf and limsup. Limits, continuity. I.1-I.4, II.1-4Notes
Sep 5,7Differentiation, Mean value theorem, Taylor SeriesIII.2-III.3,V.3 Notes
Sep 12,14Riemann IntegrationV.1-V.2Notes
Sep 19,21Normed Vector SpacesVI.1-VI.3 Notes
Sep 26,28Normed Vector Spaces, Dimensionality of SpacesVI.1-VI.5, Notes on dimensions. Notes
Oct 3,5Dimensionality of spaces, Limits in normed vector spaces and function spacesVII.1, VII.2,VII.3Notes
Oct 12Completion of SpacesVII.4Notes
Oct 17,19Open and closed Sets, Compactness VI.5, VIII.1, VIII.2, VIII.4Notes
Oct 24,25Series and Power SeriesIX.1--IX.7Notes
Oct 31, Nov 2Integral in One VariableX.1-X.7Notes
Nov 7,9Derivatives in Vector SpacesXV.1-XV.2, XVII.1-XVII.3.Notes
Nov 14,16Inverse Function TheoremXVIIINotes
Nov 21Implicit Function TheoremXVIIINotes
Nov 28, Nov 30Measure zero and content zeroX.4 AppendixNotes