Impa Summer Course 2023 - Functional Analysis
Impa Summer Course 2023 - Functional Analysis
The course will start on January 9th, and we will have a total of 17 classes of 2h. Students will receive homework and are expected to take exams.
Classes: the lectures will be streamed via youtube on the IMPA channel and will be available online. A short description of each class along with the link for its youtube video will be made available on this page.
Homework: there will be 5 sets of 10 problems, and each set will be available in this webpage in due time. The deadlines to submit solutions to each set will be posted here as well, and under normal circumstances no homework submitted after is due date will be accepted. Solutions to each set of problems will be provided only after their deadlines have passed.
Exams: There will be 4 written exams: 3 to be done in person at IMPA, and another set of problems to take home and return later. The first exam will cover the content of the first 4 lectures, the second exam will cover the content of the first 10 lectures and the third exam will cover material from the whole course. The fourth exam, will focus on the last 7 lectures.
Grades: the final evaluation of each student will be based on grading of homework and the exams.
All exams and the homework will count for 100% of the evaluation.
– Each exam are worth 25% of the grade each.
– Homework (the 5 problem sets) is worth 25% of the grade.
– The smallest value between the 4 exams and homework will be discarded when calculating the grade.
Schedule: the calendar of the course is available in the form of a Google Agenda (the one above), and anyone can subscribe by clicking here. Classes will normally take place on Monday, Wednesday and Thursday, and any changes will be posted in the calendar.
The main reference will be the first 6 chapters of Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis, supplemented with lecture notes when needed to cover any material not in Brezis's book. Many other books in functional analysis are available, and I particularly recommend:
G. Bachman, L. Narici – Functional Analysis. New York, Dover Publications Inc, 2000
D. Pellegrino, E. Teixeira, G. Botelho - Fundamentos de Análise Funcional. Rio de Janeiro, SBM, 2015
Jan 10th: Extra class - 13h at Auditorium 1.
Jan 17th: Extra class - 13h at Auditorium 1.
Jan 30th: No class.
Feb 1st: No class.
Homework: There will be 5 problem sets instead of 6. Due dates were changed and are listed below.
New due date for Problem set 3.
New due date for Problem set 4.
New due date for Problem set 5
Jan 14th: Extra class - 13h at Auditorium 1.
Change in the exam system: 4 instead of 3, and the final one will be a take home exam.
Jan 9th: Normed and Banach spaces; Inclusion, isometries and the Completion Theorem; Continuity and boundedness of linear operators; Extension of densely defined functions; Some examples of Banach and non Banach spaces; Arzelà–Ascoli Theorem; Stone–Weierstrass Theorem; Compactness and Riesz Lemma.
Jan 10th: Linear operators and the dual space; Zorn's Lemma; Hahn–Banach Theorem: Real analytic form and duality; Existence of algebraic (Hamel) basis.
Jan 11th: Hahn-Banach Theorem: Complex analytic form; Hahn–Banach Theorem: 1st and 2nd geometric forms.
Jan 12th: Phillips's Theorem for extension of operators; Bidual and orthogonality; Riesz–Markov Theorem.
Jan 16th: Baire's Category Theorem; Banach–Steinhaus Theorem; Open mapping and closed graph theorem.
Jan 17th: Projections and complement spaces; Orthogonality revisited; Unbounded operators and the Adjoint.
Jan 18th: Topology review; Weak and Weak-* topologies on a Banach space; Differences between finite and infinite dimension; Convex sets; Mazur's Theorem; Statement of Banach–Alaoglu Theorem; Tychonoff'-s Theorem.
Jan 23rd: Weak-* topology and convergence; Banach-Alaoglu Theorem; Statement of Kakutani's Theorem.
Jan 25th: Kakutani's Theorem. Helly's Lemma. Goldstine's Theorem. Metrisability and separability. Uniform convexity.
Jan 26th: Review of Lp spaces; Marcinkiewicz (diagonal) and Riesz-Thorin inteporlation Theorems.
Feb 6th: Inner product spaces; Basic properties: Cauchy-Schwarz, paralellogram law and polarization. Orthogonal projection; Orthogonal decomposition.
Feb 8th: Orthogonal systems; Complete orthonormal systems; Bessel inequality and Parseval Theorem; Fourier series; Riesz Representation.
Feb 9th: The dual of a Hilbert space; Stampacchia and Lax–Milgram Theorems; A first look into Sobolev spaces.
Feb 13th: Transpose of an operator in a Hilbert space; Weak derivatives and an intro to Sobolev spaces.
Feb 14th: Spectrum and resolvent of an operator; Decomposition of the spectrum; Compact Operators.
Feb 15th: Fredholm alternative.
Feb 16th: Spectrum of compact operators; Spectral Theorem for compact self-adjoint operators.
The course will have the following teaching assistants:
Enzo Aljovin (aljovin@impa.br)
Thyago Souza Rosa Santos (thyago.souza@impa.br)
Exercise classes: Every Wednesday on room 345, 15h-17h.
Before you submit your solutions, please read the instructions below. You should follow them to make the life of the person grading your homework easier:
Present your homework together with the homework table below filled with your name and number of the homework.
– On the column “Status”, you should fill “FS” if the the problem is fully solved, “PS” for a partially solved problem, and “NS” in case you do not submit a solution for a problem.
– In case you claim to have a partial solution, comment what you have partially solved briefly.
You can write your solutions by hand or you can use LATEX, but in any case you must submit a .pdf version of each homework by the due date.
– The .pdf version should consist of a single file containing the solutions and the table below. Merge your files in case you need. Here is a website you can use.
– Please, do not submit images (.jpg, .png etc). Convert your files to .pdf. Here is a website that does it.
– The file of homework “X” should be named “your firstname_your surname_homework_X.pdf”. Example: A student named Fulano Pessoa should submit his homework 3 file with the name “fulano_pessoa_homework_3.pdf”.
– Send your homework to mcosta@bcamath.org with a copy to each teaching assistant. Use IMPA Functional Analysis 2023 - Homework "X" of "your name" as the subject of your email.
Do not group solutions to different problems in the same page: each problem should be presented separately.
Deliver your homework in the same order given in the exercises and remember to number the pages.
Problem set 1 - Due on Jan 18th
Problem set 2 - Due on Jan 31st
Problem set 3 - Due on Feb 7th
Problem set 4 - Due on Feb 15th
Problem set 5 - Due on Feb 28th