# Course Details for A.Y. 2015/2016

#### Name:

**Analisi Funzionale (Istituzioni di Analisi Superiore mod.1) / Functional analysis**

### Basic information

##### Credits:

*:* Bachelor Degree in Mathematics 6 CFU (b)

##### Degree(s):

Bachelor Degree in Mathematics 3^{rd} anno curriculum Generale Compulsory

##### Language:

English

### Course Objectives

Learn the fundamental structures of Functional Analysis. Get familiar with the main examples of functional spaces, in particular with the theory of Hilbert spaces and Lebesgue spaces. Get familiar with the basic notions of operator theory. Be able to frame a functional equation in an abstract functional setting.

### Course Content

- Lebesgue Measure and Integration
- L^p Spaces
- Basic of Topological Vector Spaces, Normed and Banach Spaces, Linear Operators and linear functionals.
- Hilbert Spaces
- Weak topology, Weak * topology, weak compactness
- Applications of Baire Category in Functional Analysis: Uniform Boundedness, Open Mapping,
Closed Graph, Inverse Mapping.
- Banach and Hilbert adjointness, self-adjointness
- Compact Operators
- Riesz Fredholm spectral theory

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

- Understand the theory
- Be able to solve problems
- Help to choose appropriate graduate studies
- Practice mathematical reasoning, organize topics in logical order, connect theory to applications,
elaborate independent proofs. Improve unconventional thinking.
- Get the math language to study more advanced textbooks and attend research oriented courses

### Prerequisites and Learning Activities

Mathematical Analysis (not only Calculus) in one and several space variable, Linear Algebra
(including abstract Vector Speces), Set Topology (including Compact Spaces) and
Metric Spaces, Ordinary differential equations.

### Assessment Methods and Criteria

Written test.

### Textbooks

- Terence Tao, An introduction to measure theory. , American Mathematical Society, Providence, RI, ISBN: 978-0-8218-6919-2 . 2011.
* *
- Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations. , Universitext. Springer, New York,. 2011. xiv+599 pp. ISBN: 978-0-387-70913-0
* *
- Alberto Bressan, Lecture notes on functional analysis. With applications to linear partial different , Graduate Studies in Mathematics, 143. American Mathematical Society, Providence, RI,. 2013. xii+250 pp. ISBN: 978-0-8218-8771-4
* *
- Lecture notes provided by the teacher
* *
- Michael Reed, Barry Simon, Methods of modern mathematical physics. I. Functional analysis. Second edition. , Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York,. 1980. xv+400 pp. ISBN: 0-12-585050-6
* *
- Stein, Elias M.; Shakarchi, Rami , Real analysis. Measure theory, integration, and Hilbert spaces. Princeton Lectures in Analysis, III. , Princeton University Press, Princeton, NJ,. 2005. xx+402 pp. ISBN: 0-691-11386-6
* *

### Course page updates

This course page is available (with possible updates) also for the following academic years:

** To read the *** current * information on this course, if it is still available, go to the university course catalogue .
*Course information last updated on: 03 luglio 2015, 14:59*