Elements of Functional Analysis
A.Y. 2019/2020
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
Lesson period
First semester
Course syllabus
Normed spaces. Completion, completeness and absolute convergence of series, linear operators, equivalent norms, isomorphisms, bounded linear functionals, duality, separability, reflexivity, linear projections, quotient spaces. Inverse, adjoint linear operators. Characterizations of finite-dimensional normed spaces.
Classical Banach spaces. Analytic representation of bounded linear functionals. Comparison of topologies. Weierstrass-Stone and Ascoli-Arzelà theorems.
Basic theorems in Functional analysis. Hahn-Banach, Banach-Steinhaus, open mapping (closed graph) theorems and their applications. Topological complements.
Weak topologies. Topological linear spaces, locally convex spaces, topologies generated by linear functionals. Goldstine, Banach-Alaoglu and Eberlein-Smulian theorems. Metrizability.
Basics on compact operators.
Classical Banach spaces. Analytic representation of bounded linear functionals. Comparison of topologies. Weierstrass-Stone and Ascoli-Arzelà theorems.
Basic theorems in Functional analysis. Hahn-Banach, Banach-Steinhaus, open mapping (closed graph) theorems and their applications. Topological complements.
Weak topologies. Topological linear spaces, locally convex spaces, topologies generated by linear functionals. Goldstine, Banach-Alaoglu and Eberlein-Smulian theorems. Metrizability.
Basics on compact operators.
Prerequisites for admission
The contents of the courses in Mathematical Analysis 1 to 4. Basics in General Topology, in Real Analysis and in Complex Analysis.
Teaching methods
Teaching will be held by handling the matter at the board in front of the students.
Teaching Resources
Specific easily available textbooks for any subject will be indicated during classes. The following ones are highly recommended in any case.
N. Dunford, J.T. Schwartz: Linear operators, part I.
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.
N. Dunford, J.T. Schwartz: Linear operators, part I.
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
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 duration of the colloquium depends on the reaction time of the student to the proposed questions (the expected average is 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 duration of the colloquium depends on the reaction time of the student to the proposed questions (the expected average is 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:
Zanco Clemente
Shifts:
-
Professor:
Zanco Clemente