Functional Analysis - Spring Semester 2021
Part of the Hilbert Space Methods Course
Update: Due to the global state of emergency caused by COVID 19 , the IMM has suspended on-site lecturer visits until further notice and will perform the lessons of the Fall semester, 2020 through distance learning. This includes the Functional Analysis course, which is taught through a combination of pre-recorded video lectures, Google Classroom assignments and interactive teacher-to-students Q&A/exercise sessions through the Zoom video conferencing tool. These Zoom interactive sessions are also recorded and made available to the students, along with all other course materials.
Dates: February-May 2021
Faculty:
Sahibzada Waleed Noor
(Campinas, Brazil)
Alfonso Sorrentino
(Rome, Italy)
Fabrizio Bianchi
(Tutor, Lille, France)
Sahibzada Waleed Noor
Alfonso Sorrentino
Fabrizio Bianchi
Syllabus:
Norms and normed spaces; Linear maps, kernels, invariant subspaces, invertibility; Completeness and Banach spaces; Standard examples: C^0, L^p and embedding theorems; Separability; Non-compactness of the unit ball; Inner products and Hilbert spaces; Examples: l^2 L^2; Schwarz inequality; Orthogonality; Linear functional & Reisz representation; Dual spaces; Weak/Weak* convergence; Arzela-Ascoli; Basic spectral theory: compact and Hermitian operators; Some Fourier analysis: Fourier series and transforms; Some Sobolev space theory and applications: linear differential operators.
Textbooks:
Ronald G. Douglas, Banach Algebra Techniques in Operator Theory, Springer-Verlag, 1998. (Basic text for part 1)
Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, UTM-Springer, 2011. (Basic text)
Walter Rudin, Functional Analysis, McGraw-Hill, 1991. (Suplemental text)
Elias M. Stein and Rami Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis, Princeton University Press, 2012. (Additional/advanced topics)
Course Materials: coming soon.
Course Schedule (Pakistan time):
Week 1 (15/02-19/02):
Live lecture on Monday and Thursday 18:00h-19:00h
Problems class on Tuesday 14:30h-16:00h and 17:00-18:30h
Q&A session on Wednesday 14:30h-16:00h
Notes Lecture 1 - Spaces of Continuous Functions
Week 2 (22/02-26/02):
Live lecture on Monday and Thursday 18:00h-19:00h
Problems class on Tuesday 14:30h-16:00h and 17:00-18:30h
Q&A session on Wednesday 14:30h-16:00h
Notes Lecture 2 - Banach Spaces and Nets
Notes Lecture 3 - Nets and Summability
Exercise Sheet 1
Week 3 (01/03-05/03):
Live lecture on Monday and Thursday 18:00h-19:00h
Problems class on Tuesday 14:30h-16:00h and 17:00-18:30h
Q&A session on Wednesday 14:30h-16:00h
Notes Lecture 4 - Linear Functionals
Notes Lecture 5 - Some Dual Spaces
Exercise Sheet 2
Week 4 (08/03-12/03):
Live lecture on Monday and Thursday 18:00h-19:00h
Problems class on Tuesday 14:30h-16:00h and 17:00-18:30h
Q&A session on Wednesday 14:30h-16:00h
Notes Lecture 6 - Sequence and Function Spaces
Notes Lecture 7 - Compactness and Weak Topologies
Exercise Sheet 3
Week 5 (15/03-19/03):
Live lecture on Monday and Thursday 18:00h-19:00h
Problems class on Tuesday 14:30h-16:00h and 17:00-18:30h
Q&A session on Wednesday 14:30h-16:00h
Notes Lecture 8 - Axiom of choice and Alaoglu’s Theorem
Notes Lecture 9 - Hahn-Banach Theorem
Exercise Sheet 4
Week 6 (22/03-26/03):
Live lecture on Monday and Thursday 18:00h-19:00h
Problems class on Tuesday 14:30h-16:00h and 17:00-18:30h
Q&A session on Wednesday 14:30h-16:00h
Notes Lecture 10 - Universality of C(X)
Notes Lecture 11 - Quotient spaces and the Baire's theorem
Week 7 (29/03-02/04):
Midterm break and revision
Notes Lecture 12 - Opening mapping and uniform boundedness principle
Week 8 (05/04-09/04):
Midterm exam
Week 9 (12/04-16/04), beginning of part 2:
Live lecture on Monday and Thursday 18:00h-19:00h
Problems class on Tuesday 14:30h-16:00h and 17:00-18:30h
Q&A session on Wednesday 14:30h-16:00h
Notes Lecture 1
Notes Lecture 2
Exercise Sheet 1
Week 10 (19/04-23/04):
Live lecture on Monday and Thursday 18:00h-19:00h
Problems class on Tuesday 14:30h-16:00h and 17:00-18:30h
Q&A session on Wednesday 14:30h-16:00h
Notes Lecture 3
Notes Lecture 4
Notes on the Completion of a pre- Hilbert space, by Joel Feldman (University of British Columbia, Canada)
Week 11 (26/04-30/04):
Live lecture on Monday and Thursday 18:00h-19:00h
Problems class on Tuesday 14:30h-16:00h and 17:00-18:30h
Q&A session on Wednesday 14:30h-16:00h
Notes Lecture 5
Notes Lecture 6
Exercise Sheet 2
Week 12 (03/05-07/05):
Live lecture on Monday and Thursday 18:00h-19:00h
Problems class on Tuesday 14:30h-16:00h and 17:00-18:30h
Q&A session on Wednesday 14:30h-16:00h
Exercise Sheet 3
Week 13(10/05-14/05):
Holiday
Week 14 (17/05-21/05):
Live lecture on Monday and Thursday 18:00h-19:00h
Problems class on Tuesday 14:30h-16:00h and 17:00-18:30h
Q&A session on Wednesday 14:30h-16:00h
Week 15 (24/05-28/05):
Live lecture on Monday and Thursday 18:00h-19:00h
Problems class on Tuesday 14:30h-16:00h and 17:00-18:30h
Q&A session on Wednesday 14:30h-16:00h
Week 16 (31/05-04/06):
Live lecture on Monday and Thursday 18:00h-19:00h
Problems class on Tuesday 14:30h-16:00h and 17:00-18:30h
Q&A session on Wednesday 14:30h-16:00h
Week 17 (07/06-11/06):
Live lecture on Monday and Thursday 18:00h-19:00h
Problems class on Tuesday 14:30h-16:00h and 17:00-18:30h
Q&A session on Wednesday 14:30h-16:00h
Week 18 (14/06-18/06):
Live lecture on Monday and Thursday 18:00h-19:00h
Problems class on Tuesday 14:30h-16:00h and 17:00-18:30h
Q&A session on Wednesday 14:30h-16:00h
Week 19 (21/06-25/06):
Break and revision
Week 20 (28/06-02/07):
Final exam
Related Materials
ICTP Postgraduate Diploma Programme - Functional Analysis (video playlist, 2019) by Prof. Emanuel Carneiro
MTH601 - Hilbert Space Methods
Course ObjectivesThis course is intended to familiarize the students with the basic concepts, principles and methods of Hilbert spaces and its applications. The students will understand the theory of inner product spaces and their properties. After completion of this course, the students can develop and apply the basic properties of Hilbert spaces and operators on Hilbert spaces. They can also Use some advance properties of Hilbert spaces to prove some important theorems for operators.
Course Contents
Inner product spaces, completeness, orthogonality, Gram-Schmidt process, Best approximation theorem, Reisz-Frechet representation theorem, bilinear functional, linear operator theory including self-adjoint and normal operators, some applications.
Books
Ivar Stakgold, Green Functions and Boundary value problems, John Wiley and Sons, New York (1976).
L. Debnath and P. Mikusinki, Introduction to Hilbert Spaces with Applications, Academic Press, New York (1999).