# Course Details

#### Name:

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

### Basic information

##### Credits:

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

##### Degree(s):

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

##### Language:

Italian

### 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.

### Teaching Methods

**Language**: Italian

Classical traditional XX century blackboard teaching (no fancy modern technology)

### Assessment Methods and Criteria

Written test.

### Textbooks

- R.L. Wheeden A. Zygmund,
**Measure and Integral**. CRC Press,. 1977. * *
- W. Rudin,
**Real and complex analysis.**. Mc Graw Hill. . * *
- Haim Brezis,
**Functional analysis, Sobolev spaces and partial differential equations.**. Universitext. Springer, New York,. 2011. * *
- E. Kreyszig,
**Introductory Functional Analysis with applications**. Wiley. 1978. * *
- M. Reed, B.Simon,
**Methods of modern mathematical physics. I. Functional analysis. Second edition. **. Academic Press, New York,. 1980. * *

### Course page updates

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

*Course information last updated on: 18 luglio 2019, 11:16*