Marcelo Mendes Disconzi
Department of Mathematics, Vanderbilt University

email: marcelo.disconzi at vanderbilt.edu
office: Stevenson Center 1515
phone: (615) 322 7147   fax: (615) 343 0215
mail to: 1326 Stevenson Center, Vanderbilt University, Nashville TN 37240

Vanderbilt












Marcelo Mendes Disconzi
MATH 6100 - Graduate Analysis

General Information:
For a description of the course, including the grading policy, consult the course syllabus. Students are responsible for reading the syllabus and being aware of all the course and university policies.

Textbook: There will be no required textbook. See the course syllabus for a list of suggested references.

Classes meets on TR, 1:10pm–2:25pm, at Stevenson Center 1312.

Contact Information and Office Hours:
Instructor's office: SC 1515.
Instructor's email: marcelo.disconzi@vanderbilt.edu.
Instructor's office hours: Thursdays, 3-6pm, or by appointment.
Instructor's office phone: 322-7147.

General Math Office: SC 1326, phone: 322-6672.

Exams.

Description Date
Location and Time
Remarks
Test 1 Oct 13 In class
Final Exam
take-home

Schedule:
Below is a schedule for the course (for the academic calendar, click here). This will be updated regularly and, therefore, students should check this webpage frequently. The due date for each assignment will be posted as the course progresses.

Date Material covered HW problems and assignments Remarks
Aug 27 Basics.
HW1
This hw is optional; it is just to warm-up.
Sept 1 A bit of topology; compactness. HW2 HW due Sept 8
Sept 3 Topological vector spaces.

Sept 8 More TVS, convexity. HW3 HW due Sept 15
Sept 10 Convex inequalities.

Sept 15 Some functions spaces. Measures. HW4 HW due Sept 22
Sept 17 Properties of measures. Characterization in terms of inequalities.

Sept 22 Some convergence results. Real, positive, and relatively bounded measures. Ordering.

Sept 24 Student presentation: constructive definition of inductive topologies. Inequalities with positive measures. Sup and inf of measures. HW5
Sept 29 Operations with measures. Uniform convergence and uniform approximation. Lower semi-continuous functions.

Oct 1 Upper integral and outer measure. Lebesgue outer measure. HW6 Theorem proven in class
Oct 6
More on upper integral and outer measure.

Oct 8 Student presentations: properties of lower semi-continuous functions; uniform approximation; analytic Hahn-Banach from the geometric Hahn-Banach (aka first separation theorem).
Hahn-Banach theorem
Oct 13 Test.

Oct 20 Some convergence theorems; negligible sets (sets of zero measure) and functions; properties true almost everywhere. HW7
Oct 22 Equivalence classes of functions. Space of maps with finite semi-norm N_p and related properties.

Oct 27 p-integrable functions, L^p topology and L^p spaces. Completeness of L^p and other related spaces.

Oct 29 Behavior of sequence and sub-sequences. Properties of p-integrable functions. Extension of measures and integral of functions in L^1; integrable functions.

Nov 3 Upper envelope of functions in L^p. Monotone convergence theorem.

Nov 5 Dominated convergence theorem. Relation between the spaces L^p and L^q. Several remarks on integration.


Nov 5 Upper and lower envelopes of sequences of integrable functions. More convergence theorems. Integrability of lower and upper semi-continuous functions.
HW8
Make-up class for the week of Nov 16-20.
Buttrick 112 at 7pm.
Nov 10 Integrable sets and their properties.


Nov 12 Boolean rings, Boolean algebras, and sigma algebras. Step functions.


Nov 12 A Riesz representation type theorem.

Make-up class for the week of Dec 7-11.
SC 1312 at 7pm.
Nov 17 Student presentation: characterization of integrable sets; completion of metric spaces; extension of linear functionals.

Dec 1 A Riesz representation type theorem. HW9
Dec 3 A Riesz representation type theorem.

Dec 10 Last class
Last day to turn in homework.

Anonymous feedback:
Students are encouraged to bring suggestions and to discuss with the course instructor any concerns they may have, including something they think is not being properly handled in the course. But if you do not feel comfortable doing that, here you have the opportunity to send some anonymous feedback.