Analysis (Course A)
Analysis (Course A)
Academic year 2016/2017
- Course ID
- Teaching staff
- Prof. Elena Cordero
Prof. Joerg Seiler
- 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
- Written and oral
- Calculus and mathematical Analysis in one and several real variables. Ordinary Differential Equations.
Sommario del corso
This course introduces the theory of infinite-dimensional vector spaces and of the 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. The applications of this theory concern spaces of functions, and the operators (integral and differential) between them. In this course we will introduce the basic tools of modern Mathematical Analysis. This is a fundamental course for many fields of applied mathematics, in particular for Probability, Statistics and Numerical Analysis.
Results of learning outcomes
The knowledge obtained through this course consists in the fundamental mathematical tools for the understanding and the correct treatment of complex mathematical problems. The skills consist in the ability to recognize the mathematical structures and instruments used in the solution of problems concerning integral or differential equations and in the ability to solve at least the simplest cases.
Standard lectures in classroom
Learning assessment methods
The final examination consists of several questions of theory and exercises. Students will be required to know the definitions, the statements of the theorems and their applications, and to be able to solve the exercises. The final grade will be out of thirty.
- 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
- 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.
- Royden, H.L. Real Analysis. MacMillan.
Additional Lecture Notes will be made available to the students.