Functional Analysis
A.Y. 2025/2026
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
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
o Normed and Banach spaces, brief account of completion. Equivalent norms. Continuous linear operators, space of operators. Dual space. Hahn-Banach theorems. Strong compactness.
o 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.
o 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.
o Brief account of the spectral theory. Spectrum of compact operators.
o 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.
o 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.
o Brief account of the spectral theory. Spectrum of compact operators.
Prerequisites for admission
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
Frontal lectures with the use of a blackboard.
Teaching Resources
o Lecture notes (in English) of the course, by the lecturer, will be available. Further references, if any, will be indicated during the course.
o 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.
o 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
o During the semester, the student will be assigned a few homeworks of solving some exercises. The final examination consists of an oral colloquium.
o 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.
o The medium duration of the colloquium is about 45-60 minutes.
o Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral colloquium.
o 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.
o The medium duration of the colloquium is about 45-60 minutes.
o Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral colloquium.
Educational website(s)
Professor(s)