Functional Analysis
A.Y. 2026/2027
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 cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1. SPACES. Normed and Banach spaces, space of operators. Topological vector spaces (t.v.s.); seminormed spaces and their equivalence with locally convex t.v.s. Hahn-Banach theorems on extension of functionals and separation of convex sets. Strong compactness in normed spaces; Heine-Borel property in some t.v.s.
2. DUAL SPACES. Duals of some Banach spaces. General weak topologies; weak topologies of normed spaces; weak compactness and reflexivity. Metrizability of weak topologies. Annihilators.
3. CONTINUOUS LINEAR OPERATORS. Baire spaces and the Baire Category Theorem. Banach-Steinhaus theorem (Uniform Boundedness Principle). Open Mapping Theorem, Closed Graph Theorem. Adjoint operator (dual operator). Compact operators between Banach spaces. Operators on Hilbert spaces; square root of a positive operator. Integral operators with kernel: Fredholm and Volterra operators.
4. SPECTRUM. Spectrum of operators o Banach spaces: basic properties; point spectrum. Spectrum of finite-dimensional operators. Spectrum of compact operators; Fredholm theory in Hilbert spaces. Spectral decomposition of self-adjoint or normal operators on Hilbert spaces. Spectrum of Volterra operators.
2. DUAL SPACES. Duals of some Banach spaces. General weak topologies; weak topologies of normed spaces; weak compactness and reflexivity. Metrizability of weak topologies. Annihilators.
3. CONTINUOUS LINEAR OPERATORS. Baire spaces and the Baire Category Theorem. Banach-Steinhaus theorem (Uniform Boundedness Principle). Open Mapping Theorem, Closed Graph Theorem. Adjoint operator (dual operator). Compact operators between Banach spaces. Operators on Hilbert spaces; square root of a positive operator. Integral operators with kernel: Fredholm and Volterra operators.
4. SPECTRUM. Spectrum of operators o Banach spaces: basic properties; point spectrum. Spectrum of finite-dimensional operators. Spectrum of compact operators; Fredholm theory in Hilbert spaces. Spectral decomposition of self-adjoint or normal operators on Hilbert spaces. Spectrum of Volterra operators.
Prerequisites for admission
Basics of General Topology and Linear Algebra. Elements of the Set Theory, Axiom of Choice. Lebesgue measure and integral in Euclidean spaces. Definition of: normed space, Banach space, norm of a continuous linear operator, dual of a normed space. Lebesgue spaces Lp(E). Elements of Hilbert spaces. Elements of Complex function theory.
Teaching methods
Teaching will be conducted through frontal lectures.
Teaching Resources
L. Vesely, "Lecture Notes on Functional Analysis (2nd Edition)" - lecture notes by the teacher that will be provided to the students.
Additional reference/study material [NOT mandatory]:
* 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.
Additional reference/study material [NOT mandatory]:
* 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 oral exm is about 45-60 minutes.
* Final marks are given using the numerical range 0-30, and will be communicated immediately after the 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 oral exm is about 45-60 minutes.
* Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral colloquium.
MATH-03/A - Mathematical Analysis - University credits: 6
Lessons: 42 hours
Professor:
Vesely Libor
Shifts:
Turno
Professor:
Vesely LiborEducational website(s)
Professor(s)