Analysis I (Fall 2014)


Instructor: Paul E. Hand

Office: Duncan 3086

Email: hand [at]

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

Office Hours: Mondays 4-6pm


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. This course in intended to assist CAAM graduate students in preparing for the qualifying exams. See the syllabus.

Text Book: Undergraduate Analysis by Lang

Homeworks: Homework assignments will be given weekly. They will be posted on Wednesdays and will be due the following Tuesday. Two assignments will be pledged and will each count as 20% of the grade. The other homework assignments will count for 50% of the grade. Classroom participation will count as 10% of the grade. Homeworks should be handed in on time. One homework will be accepted up to a week late without consequence. In that time, you may not look at the posted solutions. Subsequent late submissions will not be accepted, unless by prior permission of the instructor or by a truly exceptional circumstance.

Classroom participation: 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 2 in classProblems. Solutions
HW 2Sep 9 in classProblems. Comments on presentations. Example of presentation notes. Solutions
HW 3Sep 16 in classProblems. Solutions
HW 4Sep 23 in classProblems Solutions
Optional HW 5Sep 30 in classProblems
Pledged HW 6Oct 7 in classProblems
HW 7Oct 21 in classProblems Solutions
HW 8Oct 28 in classProblems Solutions
HW 9Nov 4 in classProblems. Solutions
HW 10Nov 11 in classProblems Solutions
HW 11Nov 18 in classProblems Solutions
HW 12Nov 25 in classProblems Solutions
Pledged HW 13Dec 4 in classProblems

Lecture Notes

Topics and dates are tentative
Day Topics Reference (Lang) Class notes
Aug 26Real numbers.I.1-I.4, II.1Summary
Aug 28Cauchy sequences, Bolzano-Weierstrass, liminf and limsupII.1, II.2Summary.
Sep 2Limits, continuityII.2, II.4 Summary
Sep 4Squeeze theorem, limits with infinityII.2, II.3, III.1 Summary
Sep 9Differentiability, Mean value theoremIII.2, III.3 Summary
Sep 11Convex funcions, inverse function theoremIII.2, III.3Summary
Sep 16Riemann IntegrationV.1-V.2 Summary
Sep 18Riemann Integration, Taylor SeriesV.2-V.3 Summary
Sep 23Taylor Series, Normed vector spacesV.3, VI.1-VI.3 Summary
Sep 25Normed vector spacesVI.2-VI.3 Summary
Sep 30
Oct 2
Oct 7Banach spacesVI.4 Summary
Oct 9Open and closed setsVI.5 Summary
Oct 16Dimensionality of spacesNotes by Symes Summary
Oct 21Limits in normed vector spaces and function spacesVII.1, VII.2, VII.3 Summary
Oct 23Equivalence relationsVII.4 Summary
Oct 28Completion of spacesVII.4 Summary
Oct 30Sequential compactness, Compactness by open coversVIII.1, VIII.2, VIII.4 Summary
Nov 4SeriesIX.1, IX.2, IX.3 Summary
Nov 6Absolute and uniform convergenceIX.4, IX.5 Summary
Nov 11Power seriesIX.6-IX.7 Summary
Nov 13Extension of linear functionsX.1, X.2, X.3 Summary
Nov 18Integral via step functionsX.1-X.3 Summary
Nov 20Measure and contentX.4 appendix Summary
Nov 25Almost everywhere convergenceX.4 appendix Summary
Dec 2Relation of derivatives and integralsX.5-X.7 Summary
Dec 4Lebesgue IntegralX.4 appendix Summary