Analysis I (Fall 2015)


Instructor: Paul E. Hand

Office: Duncan 3086

Email: hand [at]

Lectures: TR, 10:50-12:05 in Fondren 412

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. See the syllabus.

Text Book: Undergraduate Analysis by Lang

Homeworks: This class has homework problems of two types: practice problems and portfolio problems. Practice problems are only graded for completion. Portfolio problems are problems that are similar to qualifier problems. They will be graded in a binary manner, taking into account the correctness, the elegance, and the exposition. Here are some examples of the quality of proofs expected. Portfolio problems can and should be revised until they are judged to be of professional quality. Two homework assignments will be pledged and will serve as simulated qualifier exams.

Classroom participation and in-class presentations: Each day of class, one or two students will present a theorem from that day’s content. The classroom participation grade will be based on the sincerity of your preparation for these presentations. Please follow my comments and suggested structure for the presentations.

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 1Sep 1 in classProblems. Solutions
HW 2Sep 8 in classProblems. Solutions
HW 3Sep 15 in classProblems. Solutions
HW 4Sep 22 in classProblems
Pledged HW 5Sep 29 in classProblems. Solutions. Sample from 2014
HW 6Oct 20 in classProblems.
HW 7Oct 27 in classProblems.
HW 8Nov 3 in classProblems.
HW 9Nov 17 in classProblems
HW 10Nov 24 in classProblems
Pledged HW 11Dec 1 in classProblems. Sample from 2014

Lecture Notes

Topics and dates are tentative
Day Topics Reference (Lang) Class notes
Aug 25Real numbers.I.1-I.4, II.1Summary. Notes
Aug 27Cauchy sequences, Bolzano-Weierstrass, liminf and limsupII.1, II.2Summary. Notes
Sep 1Limits, continuityII.2, II.4Summary. Notes
Sep 3Squeeze theorem, limits with infinityII.2, II.3, III.1 Summary. Notes
Sep 8Differentiability, Mean value theoremIII.2, III.3 Summary. Notes
Sep 10Convex funcions, inverse function theoremIII.2, III.3Summary. Notes
Sep 15Riemann IntegrationV.1-V.2Summary. Notes
Sep 17Riemann IntegrationV.2Summary. Notes
Sep 22Taylor SeriesVI.3Summary. Notes
Sep 24Slack
Sep 29Slack
Oct 1NormsVI.1-VI.3Summary. Notes
Oct 6Inner products and norm equivalenceVI.1-VI.3 Summary. Notes
Oct 8Complete normed vector spacesVI.4Summary. Notes
Oct 15Open and closed setsVI.5 Summary. Notes
Oct 20Dimensionality of spacesNotes by SymesSummary.
Oct 22Limits in normed vector spaces and function spacesVII.1, VII.2, VII.3Summary. Notes
Oct 27Equivalence relationsVII.4Summary. Notes
Oct 29Completion of spacesVII.4Summary. Notes
Nov 3Sequential compactness, Compactness by open coversVIII.1, VIII.2, VIII.4Summary. Notes
Nov 10SeriesIX.1, IX.2, IX.3, IX.4, IX.5Summary. Notes
Nov 12Power seriesIX.6-IX.7Summary. Notes
Nov 17Extension of linear functionsX.1, X.2, X.3Summary
Nov 19Integral via step functionsX.1-X.3Summary
Nov 24Measure and contentX.4 appendixSummary
Dec 1Almost everywhere convergenceX.4 appendixSummary
Dec 3Relation of derivatives and integralsX.5-X.7Summary