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## ELEMENTS OF ADVANCED ANALYSIS 2

CODE 61705 2018/2019 8 credits during the 1st year of 9011 Mathematics (LM-40) GENOVA MAT/05 Italian GENOVA (Mathematics) 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 "Istituzioni di Analisi Superiore 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 "IAS 1".

### TEACHING METHODS

Teaching style: lectures, in presence (72 hours). Each week, one hour 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: spectrum, adjoint operator, compact operators, the spectral theorem for compact selfadjoint operators, unbounded operators, closed operators; elements of spectral theory in Banach algebras; Gelfand theory for commutative Banach algebras; introduction to C*-algebras; the Gelfand representation theorem for commutative C*-algebras.

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

W. Rudin: Real and Complex Analysis

W. Rudin: Functional Analysis

## TEACHERS AND EXAM BOARD

### Exam Board

LAURA BURLANDO (President)

GIANFRANCO BOTTARO

## LESSONS

### TEACHING METHODS

Teaching style: lectures, in presence (72 hours). Each week, one hour will be devoted to exercises.

### LESSONS START

The class will start according to the academic calendar.

## 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 counsult 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.

### Exam schedule

Date Time Location Type Notes
25/06/2019 10:00 GENOVA Scritto
23/07/2019 10:00 GENOVA Scritto
10/09/2019 10:00 GENOVA Scritto

### FURTHER INFORMATION

Attendance: recommended.

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