Mathematical Analysis - Fall Semester 2020
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 Mathematical 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: 15/09/2020 - 31/01/2021
Faculty:
Emanuel Carneiro (Trieste, Italy)
Emanuel Carneiro
Jose M. Conde Alonso (Madrid, Spain)
Jose M Conde
Alexandra Otiman
(Tutor, Rome, Italy)
Alexandra Otiman
Fabrizio Bianchi
(Tutor, Lille, France)
Fabrizio Bianchi
Syllabus:
Professor Carneiro
Basic concepts of measure theory (Sigma algebras, Measurable functions, Measures). Integration (Simple functions, Non-negative functions, The class of integrable functions). The Lebesgue measure on Rd (Preliminaries on rectangles and cubes, The exterior measure, The Lebesgue measurable sets and the Lebesgue measure). Constructing general measures (Outer measures, Charateodory's theorem, Pre-measures, Borel measures on R).
Professor Conde Alonso
Review: measures, integration, convergence theorems, Lebesgue measure. Other examples of measures. Lp spaces I: de finition, triangle inequality of the norm, Hölder's inequality, completeness. Lp spaces II: Jensen's inequality, dual space. Hilbert spaces: Riesz representation theorem. Differentiation of measures and Radon-Nykodim theorem. Fubini's theorem. Maximal functions I: Hardy-Littlewood maximal function, dyadic maximal function, cubes and balls. Maximal functions II: Chebyshev's inequality, Lorentz spaces and interpolation. Spaces of finite measure: conditional expectations, martingales, martingale inequalities, spaces of martingales.
Textbooks:
R.G. Bartle, The Elements of Integration and Lebesgue Measure, John Wiley and Sons, 1966 (main textbook for Part 1) [Bart]
E.M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration and Hilbert Spaces, Princeton Lectures in Analysis, Book 3. Princeton University Press, 2009.
G. Folland, Real Analysis - modern techniques and their applications. Wiley-Interscience. (Lp spaces, Hilbert spaces, differentiation of measures, Fubini's theorem) [Foll]
J. Duoandikoetxea, Fourier Analysis. Graduate Studies in Mathematics, vol. 29, AMS. (maximal functions)
R. Durrett, Probability: theory and examples. Cambridge University Press. (martingales)
Course Materials:
ICTP Postgraduate Diploma Programme - Real Analysis (video playlist, 2021) by Prof. Emanuel Carneiro
Course Schedule (Pakistan time):
Week 1 (05/10-09/10):
Watch Lecture 1 (Basic concepts of Measure Theory)
Week 2 (12/10-16/10):
Watch Lecture 2 (Measures) by 14/10
Whiteboard notes Lecture 3
Whiteboard notes Lecture 4
Exercise sheet 1 - due on 14/10, 14:30
Week 3 (19/10-23/10):
Read chapters 4 and 5 of [Bart] and solve the exercises
Watch Lecture 3 (Integration) by 19/10
Whiteboard notes Lecture 5
Whiteboard notes Lecture 6
Exercise sheet 2 - due on 21/10, 23:59
Week 4 (26/10-30/10):
Watch Lecture 4 and Lecture 5 (Lebesgue measure)
Whiteboard notes Lecture 7
Whiteboard notes Lecture 8
Exercise sheet 3 - due on 28/10, 23:59
Week 5 (02/11-06/11):
Whiteboard notes Lecture 9
Exercise sheet 4
Week 6 (09/11-13/11):
Midterm exam
Week 7 (16/11-20/11):
Break in lessons (preparation for the midterm exam)
Week 8 (23/11-27/11), beginning of part 2:
Live lectures on Wednesday 16:30h-18:00h and Thursday 14:30h-16:00h
After the Wednesday lecture, Watch Lecture 7 (Littlewood’s Principles, Lp spaces)
Read chapters 6 and 7 of [Bart] and chapter 6 of [Foll]
Class notes Lecture 1
Class notes Lecture 2
Problem set 1
Week 9 (30/11-4/12):
Live lectures on Monday 13:30h-15:00h and Thursday 14:30h-16:00h
Watch Lecture 8 (Lp spaces)
Class notes Lecture 3
Week 10 (7/12-11/12):
Live lectures on Monday 13:30h-15:00h and Thursday 14:30h-16:00h
Watch Lecture 9 (Lp spaces, Jensen’s inequality, Hilbert spaces)
Class notes Lecture 4
Problem set 2
Week 11 (14/12-19/12):
Live lectures on Monday 13:30h-15:00h and Thursday 14:30h-16:00h
Lp spaces - Some useful inequalities
Dual spaces - Linear functionals
Dual spaces - Hilbert spaces
Dual spaces - Differentiation of measures
Dual spaces - The dual of Lp
Problem set 3
Week 12 (21/12-25/12):
Live lecture on Monday 13:30h-15:00h
Related Materials
Mathematical Analysis Preparatory Sessions:
Week 1:
Warm-up Questions
Lecture 1: Sequences
Lecture Notes 1
Whiteboard Notes 1
Problem Sheet 1
Lecture 2: Bolzano–Weierstrass Theorem
Whiteboard Notes 2
Problem Sheet 2
Week 2:
Lecture 3: Topology Rd, Uniform Continuity
Whiteboard Notes 3
Lecture 4: Uniform Continuity, Pointwise and Uniform Convergence
Whiteboard Notes 4
Lecture 5
Whiteboard Notes 5
ICTP Diploma Programme - Real Analysis (video playlist) by Prof. Stefano Bianchini
MTH622 - Mathematical Analysis
Course ContentsBooks