Elements of Functional Analysis | Università degli Studi di Milano Statale

A.Y. 2024/2025

Max ECTS

Overall hours

SSD

MAT/05

Language

Italian

Included in the following degree programmes

Mathematics (Classe LM-40)-Enrolled from 2012/13

Learning objectives

The aim of the course is to provide basic notions and tools in the (infinite-dimensional) setting of Linear Functional Analysis. The course is devoted to supply background for advanced courses.

Expected learning outcomes

Knowledge of the Functional Analysis basic techniques and their use in solving simple theoretical problems as well as simple problems in Applied Mathematics.

Lesson period: First semester

Lessons timetable

Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi

Exams calendar

Single course

This course can be attended as a single course.

Take a single course

Course syllabus and organization

Single session

Responsible

Peloso Marco Maria

Lesson period

First semester

Syllabus
(active tab)
Course structure

Course syllabus

Normed and Banach spaces, brief account of completion. Equivalent norms. Continuous linear operators, space of operators. Dual space. Hahn-Banach theorems. Strong compactness.

Examples of Banach (function and sequence) spaces and their duals. Topological vector spaces. Weak topologies, reflexivity, weak and weak-star compactness. Brief account of metrizability of weak topologies.

Baire's Category Theorem and its applications: Uniform Boundedness Principle, Open Mapping Theorem and Closed Graph Theorem. Adjoint operator. Compact operators. Fredholm and Volterra integral operators.

Brief account of the spectral theory. Spectral theory of compact operators.

Prerequisites for admission

The contents of the courses in Mathematical Analysis 1 to 4. Basics in General Topology, in Linear Algebra, and in Real Analysis, in particular is fundamental the familiarity with L^p and Hilbert spaces.

Teaching methods

The course is offered as standard blackboard lectures. The solution of assigned homework will be required.

Teaching Resources

-- Note del corso
-- John B. Conway, A Course in Functional Analysis 2nd Ed., Springer New York, NY, 2007.
-- [Ru] W. Rudin, Functional Analysis, 3rd Edition, McGraw-Hill, New York, 1991.

Assessment methods and Criteria

During the semester, the student will be assigned some homework consisting of some exercises. The final examination consists of an oral examination. The student will be required to illustrate and to discuss results presented during the course or directly connected with them, as well as to solve problems in that context, in order to evaluate her/his knowledge and comprehension of the subjects covered as well as the ability in connecting and applying them correctly. The typical duration of the oral exam is about 45 minutes.

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6

Lessons: 42 hours

Professor: Peloso Marco Maria

Shifts:

Turno

Professor: Peloso Marco Maria

Educational website(s)

Elementi di Analisi Funzionale (a.a. 2024/25)

Professor(s)

Peloso Marco Maria

Web site

Reception:

By appointment

My office, room 1021 Dipartimento di Matematica

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