ANALYSIS (Course B)
Academic year 2015/2016
- Course ID
- Bertrand Lods
- 1st year
- Teaching period
- First semester
- D.M. 270 TAF B - Distinctive
- Course disciplinary sector (SSD)
- MAT/05 - analisi matematica
- Class Lecture
- Type of examination
- A good knowledge of basic calculus and real analysis.
Sommario del corso
This course is a 9-credit course aimed at introducing and developing many of the mathematical tools necessary in many fields of Probablity, statistics and applied mathematics. It introduces in particular the theory of infinite-dimensional vector spaces with a special focus on the concepts of normed vector spaces, completeness, compactness, and other characteristic properties of infinite dimensional vector spaces. Concrete applications to spaces of functions will be provided. Application of this theory to the study of ordinary differential equation and Fourier analysis will be also given.
Results of learning outcomes
At the end of the course, the student is expected to be capable of:
- using the basic tools and results to pose, formalize and solve a complex mathematical problem of applied interest.
- being able to think about possible and useful generalizations of the various results studied during the lectures.
- being able to communicate such findings using appropriate and clear mathematical notation and language
The course is articulated in 72 hours of formal in‐class lecture time, and in at least as many hours of at‐home work solving practical exercises.
Learning assessment methods
The course grade is determined solely on the basis of a written examination. The examination (2 hours and 45 minutes) test the student's ability to do the following:
Present briefly the main ideas, concepts and results developed in the course, also explaining intuitively the meaning and scope of the definitions and the arguments behind the validity of the result. Students will be required to know the definitions, the statements of the theorems, the idea behing the proofs and their applications.
Use effectively the concepts and the result to answer questions pertaining to functional analysis.
The above is accomplished by asking the student to answer 5‐6 questions. Each of the questions has an essay part, and some of the questions also have a more practical ("exercise ") part.
The course includes exercises classes; extra exercises are suggested as homework.
ProgramThe course is divided in 4 parts:
- Reminders of multivariate calculus
- Functions of several variables, differential calculus;
- Linear algebra: matrices, determinants, diagonalization
- Series of numbers and series of functions;
- Integral calculus for functions of several variables.
- Abstract vector spaces
- Topological metric spaces, compactness;
- Metric spaces: properties, continuity of functions;
- Banach spaces: fundamental properties, examples;
- Hilbert spaces; fundamental properties, projection theorem.
- L_p spaces.
- The space of continuous functions, Ascoli-Arzela Theorem.
- Ordinary Differential Equations
- Cauchy Lipschitz theory;
- Classical methods of integration of ODE's (linear equations, Laplace transform, separation of variables).
- Fourier Analysis.
- Fourier series, Fourier transform;
Suggested readings and bibliography
KHURI, A.I. Advanced calculus with applications in statistics. Wiley Series in Probability and Statistics.
- ROYDEN, H.L. Real Analysis. MacMillan.
- RYNNE, B. P. and Martin A. YOUNGSON, M.A., Linear Functional Analysis, Second Edition, Springer 2008
- DUDLEY, R. M., Real Analysis and Probability, Cambridge University Press.
- Additional Lecture Notes will be made available to the students.
Days Time Classroom