Analysis I (Fall 2016)


Instructor: Paul E. Hand

Office: Duncan 3086

Email: hand [at]

Lectures: TR, 10:50-12:05 in BCN 146

Office Hours: Mondays 4-6pm


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 30 in classProblems.
HW 2Sep 6 in classProblems.
HW 3Sep 13 in classProblems.
HW 4Sep 27 in classProblems.
HW 5Oct 4 in classProblems.
Pledged HW 6Oct 18 in classProblems. Practice Exam 2015 with solutions. Practice Exam 2014.
HW 7Oct 25 in classProblems.
HW 8Nov 1 in classProblems
HW 9Nov 8 in classProblems.
HW 10Nov 15 in classProblems.
Pledged HW 11Dec 8 by noonProblems. Final from 2014. Final from 2015

Lecture Notes

Topics and dates are tentative
Day Topics Reference (Lang) Class notes
Aug 23,25Real numbers. Cauchy sequences, Bolzano-Weierstrass, liminf and limsup. Limits, continuity. I.1-I.4, II.1-4Notes
Aug 30,Sep 1Differentiation, Mean value theorem, Taylor SeriesIII.2-III.3,V.3 Notes
Sep 6,8Riemann IntegrationV.1-V.2Notes
Sep 13,15Normed Vector SpacesVI.1-VI.3 Notes
Sep 20,22Normed Vector Spaces, Dimensionality of SpacesVI.1-VI.5, Notes on dimensions. Notes
Sep 27,29Dimensionality of spaces, Limits in normed vector spaces and function spacesVII.1, VII.2,VII.3Notes
Oct 4,6Completion of Spaces, Open and closed SetsVII.4, VI.5Notes
Oct 13CompactnessVIII.1, VIII.2, VIII.4Notes
Oct 18,20Series and Power SeriesIX.1--IX.7Notes
Oct 25,27Integral in One VariableX.1-X.7Notes
Nov 1,3Derivatives in Vector SpacesXV.1-XV.2, XVII.1-XVII.3.Notes
Nov 8,10Inverse Function TheoremXVIIINotes
Nov 15,17Implicit Function TheoremXVIIINotes
Nov 22Measure zero and content zeroX.4 AppendixNotes
Nov 29, Dec 1Multiple IntegralsXX