OVERVIEW

The lectures are delivered in Italian.

This course deals with important topics in Functional Analysis and Operator Theory, which complete and develop the ones already covered in the previous course of Functional Analysis 1.

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

To provide some fundamental contents in Functional Analysis and Operator Theory.

Expected learning outcomes:

The students will become acquainted with the concepts and proofs carried out in class and how they are used in practice to solve exercises; moreover they will know how to produce easy variants of demonstrations seen and construct examples on topics covered in this course.

PREREQUISITES

Mathematical Analysis I, 2, 3 and 4, Linear Algebra and Analitic Geometry, Geometry 1, Functional Analysis 1.

TEACHING METHODS

Lectures (52 hours). Both theory and exercises are presented by the teacher in the usual way. Moreover some tutorial exercitations will be carried out during the semester.

SYLLABUS/CONTENT

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.

RECOMMENDED READING/BIBLIOGRAPHY

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

N. Dunford, J.T. Schwartz - Linear Operators. Part I: General Theory - Interscience 1957

W. Rudin - Real and Complex Analysis - McGraw-Hill 1970

W. Rudin: Functional Analysis

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Written and oral tests. Oral test, which students can access whatever the outcome of written test, has to be taken in the same exam session of written test.

If the University introduces again the obligation to perform even partially online exams (as happened in part of the academic years 2019-20, 2020-21 and 2021-22), the exam will consist only of oral test.

Students with a certified DSA ("specific learning disabilities"), disability or other special educational needs are advised to contact the lecturer at the beginning of the course in order to agree on teaching and examination methods that, while respecting the teaching objectives, take into account individual learning methods and provide suitable compensatory tools.

ASSESSMENT METHODS

The written examination consists in some exercises about the topics covered in this course. In this test the ability to apply theoretical results in concrete situations is evaluated.

The test lasts two hours and it is possible to consult the notes and textbooks.

In the oral exam a discussion of the written examination is done; moreover some questions are asked on the course content and/or about the solution of some exercises about the topics covered in this course. In such way they are assessed the understanding, the knowledge of the concepts, and the skills in using them, acquired by the students.

FURTHER INFORMATION

Attendance is recommended.

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