Advanced Analysis

Spring 2011

Instructor: Antar Bandyopadhyay (Email: antar (at) isid (dot) ac (dot) in Office: 208 Faculty Building).

Class Time: Tue Thu 14:00 - 16:00 in Class Room 22.

Instructor's Office Hours: Tue Thu 16:00 - 17:30

Course Duration: January 10 - April 22, 2011 (Note: During February 8 - 17, 2011, Professor Rajendra Bhatia will teach Fourier Series).

Midterm Examination: March 4, 2011 (Friday) 10:00 - 12:00 hours.

Final Examination: April 27, 2011 (Wednesday) 10:00 - 13:00 hours.

Course Outline:

Topological Preliminaries:
- Definition of a topological space.
- Definitions and basic properties of compact, Hausdorff, locally compact spaces.
- Definition and basic properties of continuous functions.
- Continuous image of compact set is compact.
- Urysohn's Lemma and Partition of Unity.
Positive Borel Measures:
- Regularity properties of Borel Measures.
- Continuity properties of measurable functions.
- The Riesz Representation Theorem for positive linear functionals (without proof).
- The Lebesgue measure.
Fourier Series: (taught by Professor Rajendra Bhatia, February 8 - 17, 2011)
- Definition of trignometric series.
- Abel and Cesaro summability.
- Dirichlet and Fejer kernals.
- Pointwise convergence and divergence.
- Some applications.
L_p-Spaces:
- Definitions and basic properties.
- Inequities (Jensen, Holder, Minkowski).
- Completeness.
- Approximation by continuous functions.
- Riesz Representation Theorem for L_p-Spaces (bounded linear functionals and dual space).
Signed Measures:
- Definition of signed measures.
- Positive and negative sets.
- Hahn-Jordan Decomposition.
- Absolute continuity of two σ-finite measures.
- Radon-Nikodyme Theorem.
- Lebesgue Decomposition.
Conditional Expectation:
- Definition, examples and special cases.
- Properties of conditional expectation, linearity, order-preserving, idempotent and constant-preserving.
- MCT, Fatou's Lemma and DCT for conditional expectations.
- Jensen's Inequality.
- Conditional expectation as projection on L₂.
- Conditional expectation and independence.
Discrete Parametr Martingale Theory:
- Definition of filtration and adopted sequence.
- Martingales, definitions and examples of discrete parameter sub-martingale, martingale, super-martingale.
- Basic properties.
- Doob's Maximal Inequality, Kolmogorov's Maximal Inequality.
- Definition of stopping time, stopped process, stopped σ-algebra, examples and properties.
- Predictable processes, Discrete martingale transform.
- Concept of upcrossings. Doob's upcrossing inequality.
- The (sub) Martingale Convergence Theorem, Convergence theorem for non-negative super-martingale.
- Backward Martingales, applications to proof of SLLN for i.i.d. random variables.

References:

For Tpological Preliminaries:
- Real and Complex Analysis by Walter Rudin.
- Topology by James R. Munkres.
- Topology and Mordern Analysis by G. F. Simmons.
For Positive Borel Measures, L_p-Spaces, Riesz Representation Theorem:

Real and Complex Analysis by Walter Rudin.
Real Analysis by H. L. Royden.
Functional Analysis by Walter Rudin.
For Fourier Series:

Fourier Series (Second Edition) by Rajendra Bhatia.
Fourier Series by G. H. Hardy and W. W. Rogosinski.
An Introduction to Harmonic Analysis by Yitzhak Katz nelson.
For Signed Measures:

Probability and Measures by P. Billingsley.
Real Analysis by H. L. Royden.
For Conditional Expectations and Martingale Theory:

Probability and Measures by P. Billingsley.
Probability Theory and Examples by R. Durrett.
Foundations of Calculus of Probability by J. Nevue.
Discrete Parameter Martingales by J. Nevue.

Except for the Fourier Series part, we will mostly follow Real and Complex Analysis by Walter Rudin. For Fourier Series we will follow Fourier Series by G. H. Hardy and W. W. Rogosinski. Only occasionally the others may be needed. Though they are all very good and important sources to learn measure theory/probability theory.

Prerequisites:

Real Analysis (at the level of Principles of Mathematical Analysis by W. Rudin).
Liner Algebra (at the level of Finite Dimensional Vector Spaces by P. R. Halmos).
Basic Measure Theory (at the level of Probability and Measure by P. Billingsley).

Grading Policy:

Assignments: 20% of the total credit.
Quizzes: 10% of the total credit.
Midterm Exam: 20% of the total credit.
Final Exam: 50% of the total credit.

Assignment Policies:

There will be a total of 6 sets of homework assignments each carrying a total of 40 points. 5 best assignment scores will be taken for the final grading.
The assignments will be given in class on every alternate Tuesday, and it will be due in class on Friday of the following week. Each assignment set will be based on course materials covered in the lectures in the week before and the week it is assigned. For example, the assignment given on January 18, 2011 (Tuesday) will be due on January 28, 2011 (Friday), and will be on materials covered during January 11 - 20, 2011.
Late submission of an assignment will not be accepted. If you can not submit an assignment on time, don't worry about it, and try to do well in the others. It will not count in your final grade since you have few extra assignments anyway.
Graded assignments will be returned in the class on the next Tuesday following the due date. For example, the assignment which is due on January 28, 2011 (Friday) will be returned after grading on February 1, 2011 (Tuesday).
Click here for downloading the assignments.

Quiz Policies:

There will be exactly two quizzes as surprise tests given in the class. This means there shall be no pre-scheduling. A quiz will cover all the materials done in the lectures prior to it.
There will be NO supplementary quiz given for any student who may miss a quiz for whatsoever reason. If you miss one then do not worry try doing well in the other.
Each quiz will be a closed books, closed notes test.
Quiz # 1.
Quiz # 2.

Exam Policies:

Each examination (except for the quizzes) will be an open note examination. That means, students are allowed to bring his/her own hand written notes, study materials, list of theorems etc.

Last modified April 27, 2011.