Elements of Functional Analysis
A.Y. 2024/2025
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
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.
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.
-- 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 MariaEducational website(s)
Professor(s)
Reception:
By appointment
My office, room 1021 Dipartimento di Matematica