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## FUNCTIONAL ANALYSIS 2

CODE 61705 2022/2023 6 cfu during the 1st year of 9011 MATEMATICA(LM-40) - GENOVA MAT/05 GENOVA 2° Semester AULAWEB

## OVERVIEW

This course deals with impotant topics in Functional Analysis and Operator Theory, which complete and develop the contents of the course "Analisi Funzionale 1".

The lectures are delivered in Italian.

## AIMS AND CONTENT

### LEARNING OUTCOMES

To provide some fundamental contents in Functional Analysis and Operator Theory that are considered important for the students who want to follow a PhD, or anyway aim to get a well grounded knowledge in the basic branches of Mathematics.

### AIMS AND LEARNING OUTCOMES

This course provides some fundamental contents in Functional Analysis and Operator Theory. Students are expected to understand the concepts and the proofs that have been shown in the classes. In particular, they are expected to be able to provide proofs consisting in simple variants of the proofs that have been shown in the classes, to construct examples and to solve exercises about the topics developed in the course.

### PREREQUISITES

Prerequisites: the contents of the course "AF 1".

### TEACHING METHODS

Teaching style: lectures (52 hours), given in the mode that will be decided by the University of Genova. Some of the lectures will be devoted to exercises.

### SYLLABUS/CONTENT

Quotient normed spaces; reflexivity; weak topologies; compactness in spaces of continuous functions: the Ascoli-Arzelà theorem; elements of operator theory: the spectrum, the adjoint operator, compact operators, the spectral theorem for compact selfadjoint operators, the spectral radius.

A. E. Taylor, D. C. Lay: Introduction to Functional Analysis, 2nd edition

W. Rudin: Real and Complex Analysis

W. Rudin: Functional Analysis

## LESSONS

### LESSONS START

The class will start according to the academic calendar.

### Class schedule

FUNCTIONAL ANALYSIS 2

## EXAMS

### EXAM DESCRIPTION

Written and oral tests. The written test lasts two hours and consists of two exercises concerning the contents of the syllabus. During the written test, students are allowed to consult the notes they have taken in the classes, as well as textbooks. The oral test can be taken whichever grade the student has got in the written test, and must be taken in the same round of the written test.

### ASSESSMENT METHODS

There will be both a written test and an oral one. Each of them is intended to verify understanding by the student of the concepts and of the proofs developed in the classes, as well as ability in producing proofs consisting of simple variants of the proofs that have been shown in the classes, in constructing examples and in solving exercises concerning the syllabus.

### FURTHER INFORMATION

Students are suggested to enroll in AulaWeb, in order to be able to get further information by the teachers about the course.