Elements of Functional Analysis

A.Y. 2020/2021
6
Max ECTS
42
Overall hours
SSD
MAT/05
Language
Italian
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.
Course syllabus and organization

Single session

Responsible
Lesson period
First semester
All information on the course and related material (lecture notes, lesson recordings, homework, ...) will be published on the Ariel platform.

The lectures will be delivered synchronously online through the Zoom meetings platform. They will also be recorded, but it is strongly recommended to follow them live.

The homework will be assigned on Ariel, its solutions must be sent by email to the teacher.

Oral exams will be carried out on Zoom or in presence, by appointment.
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. Brief account of 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. Spectrum of compact operators. Spectrum of compact self-adjoint operators in Hilbert spaces.
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 (L_p spaces, Hilbert spaces).
Teaching methods
Teaching will be held by handling the matter at the board in front of the students, or online synchronously, according to decision of the University.
Teaching Resources
Lecture notes (in English) of the course, by the lecturer, will be available. Further references, if any, will be indicated during the course.
In any case, the following are recommended for consultation:
- M. Fabian, P. Habala, P. Hajek, V. Montesinos, V. Zizler: Banach Space Theory, CMS Books in Mathematics, Springer.
- R. Megginson: An introduction to Banach space theory, Springer.
- W. Rudin: Real and complex analysis, McGraw-Hill.
- W. Rudin: Functional Analysis, McGraw-Hill.
Assessment methods and Criteria
During the semester, the student will be assigned a few homeworks of solving some exercises. The final examination consists of an oral colloquium.
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 medium duration of the colloquium is about 45-60 minutes.
Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral colloquium.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor: Vesely Libor
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Professor(s)