Analysis (Course A)
Analysis (Course A)
Academic year 2020/2021
- Course ID
- Teaching staff
- Prof. Elena Cordero
Albert James Milani (Lecturer)
- 1st year
- Teaching period
- First semester
- D.M. 270 TAF B - Distinctive
- Course disciplinary sector (SSD)
- MAT/05 - analisi matematica
- Formal authority
- Type of examination
- Written and oral
- Calculus and mathematical Analysis in one and several real variables. Ordinary Differential Equations.
Sommario del corso
The course introduces the participants to the theory of infinite-dimensional vector spaces and of linear operators between them, with a special focus on the concepts of normed vector spaces, completeness, compactness, and the different topologies which characterize the infinite dimensional vector spaces. Applications concern various spaces of functions and operators between them (in particular, integral and differential operators). The course presents basic tools of modern mathematical analysis which are of fundamental importance in many branches of pure and applied mathematics, in particular in probability theory, statistics, numerical analysis, partial differential equations and dynamical systems.
Results of learning outcomes
- Knowledge and understanding
The student will acquire knowledge and understanding of many basic tools which are of common use in the analysis of infinite dimensional vector spaces. In particular he will learn the theory of Banach and Hilbert spaces and their dual spaces, of linear, bounded, and compact operators, and he will know the theory of distributions (generalized functions), as well as the Fourier and Laplace transform.
- Applying knowledge and understanding
He will be able to solve simple problems and exercises related to the theory (in particular, to solve simple integral or differential equations) and he will be able to rigorously prove main results of the theory.
- Making judgements
The student will be able to select the appropriate method to solve problems and exercises related to the theory.
- Communication skills
Students will properly use mathematical language to prove the theorems and solve exercises related to the theory.
- Learning skillsThe students will acquire critical thinking abilities, as well as capacities of collaborating. These skills will help the students to improve their learning capacities.
The course is articulated in 72 hours of formal in‐class lecture time, and in at least 150 hours of at‐home work solving practical exercises.
Warning: Due to the COVID-19 emergency, lessons will take place mainly in remote mode.
The new webex links to lectures will be posted on the moodle page of the course every week.
Learning assessment methods
During the Covid-19 emergency the learning assessment method will consist of a written exam with video surveillance on Webex.
The assessment consists in a written test followed by an oral examination, after completion of the course.
The written test consists in open questions and exercises on the topics treated in class and has a duration of 180 minutes. The mark will be expressed in thirtieth; the single points (30 in total) will be distributed to the questions and exercises on the basis of their importance and length; the final score will be given by summing up the partial scores of each question and exercise.
The oral examination is scheduled after the written test and can be given only after having passed the written test with a mark of 18 or better. The oral examination consists of questions on the written test and on the topics treated in class and listed in the examination programme (which is available to the participants on the web-site of the course).
Both written test and oral examination will result in a final mark expressed in thirtieth; the minimal mark allowed for successful assessment is 18. Otherwise, the student's performance is considered insufficient and the student has to repeat the examination (both written test and oral examination).
Both written test and oral examination must be achieved in the same examination period.
The final grade will be a combination of the written test grades (90%) and the oral exam grade (10%).
- Banach spaces.
- Linear operators.
- Hilbert spaces, projections, orthonormal basis.
- Generalized Fourier series.
- Dual spaces: linear functionals, weak convergence.
- Compactness in finite dimensional spaces.
- Compact operators and applications to integral equations.
- Fundamentals of spectral theories
- Distributions (generalized functions)
- Fourier transform
- Laplace transform
Suggested readings and bibliography
- Bryan P. Rynne and Martin A. Youngson, Linear Functional Analysis, Second Edition, Springer, 2008.
- Dudley, R.M., Real Analysis and Probability, Cambridge University Press.
- Hörmander, L., The Analysis of Linear Partial Differential Operators I, Distribution Theory and Fourier Analysis, Springer, 2003.
- Royden, H.L., Real Analysis, MacMillan.
- Rudin, W., Functional Analysis, McGraw-Hill.
- Rudin, W., Real and Complex Analysis, McGraw-Hill.
Additional Lecture Notes will be made available to the students.