Functional Analysis

This is an introductory Functional Analysis course at the PhD level. A basic outline of the syllabus for the

course can be found at this link under Math 824 .

Schedule: the class meets twice a week, Monday/Thursday 3.30 pm - 4:55 pm, and extra problem sessions

(to be determined)

References:

There is no one book that will be linearly followed. However, almost all of the material covered in class

should be found in some guise in one of the following standard references:

1. P. Lax, Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience, John Wiley &

Sons, New York, 2002. xx+580 pp.

2. Michael E. Taylor, Partial differential equations I. Basic theory. Second Edition, Applied Mathematical

Sciences, 115. Springer, New York, 2011. xxii+654 pp.

3. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol I - IV, Elsevier 1981.

Notes (to be updated after each class).

Presentations: Gelfand-Mazur theorem (thanks to Saikat)

Homework 1: Exercises 1.2, 1.4, 1.6, 1.7, 1.8, 1.12, 1.16, 1.18, (first part of) 1.19, 1.21, 1.23,

1.27, 1.28, 1.32, 1.33, 1.34.

Homework 2: Exercises 3.1, 3.2, 3.4, 3.7, 3.8, 3.9, 3.12, 4.7, 4.8, 4.9, 4.10, 4.11, 4.12, 4.13.

Homework 3: Exercises 5.2, 5.3, 5.5, 5.6, 5.8, 5.11, 5.14, 6.2, 6.3, 6.4, 6.5, 6.6, 6.8, 6.10,

6.14, 6.18.

Homework 4: Exercises 8.5, 8.6, 8.7, 8.21, 8.22, 9.8, Proposition 9.5, 11.4, 12.3,

Miscellaneous Problem Set

Grading policy: The midsem exam will be based on presentations, and the final exam will be oral interviews.

Possible topics for presentation: Krein-Milman (Manoj), Schauder fixed point theorem (Hari Prasadh), Stone-

Weierstrass (Saad), Gelfand-Mazur (Saikat), Lax-Milgram, Hille-Yosida/semigroups (Soham), local version of Malgrange-

Ehrenpreis, RKHS and representer theorem (Siddhartha), Kato-Rellich and selected applications (Manish), Fredholm alternative

and elementary properties of spectra of compact operators (Sakshi), index of Fredholm operators (Akshaya), application

of Kato-Rellich to Schroedinger operators (including Coulomb potential) (Soumyajit), Mountain pass lemma (Dharamveer).

Contact information:

Email: mathmukherjee@gmail.com or mukherjee@math.iitb.ac.in

Office: B1-B, Math Department (in the basement)

General advice: It is highly recommended that you consult other standard texts in the field, to appreciate

different tastes in selection of materials, or differing slants on the same material. For example, even in the

three references listed above, this difference is clearly perceptible.