Elements of Functional Analysis
A.Y. 2018/2019
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. Subjects among the following ones will be developed, omitting those the audience would be familiar with.
Expected learning outcomes
Knowledge of the topics of the course and their applications.
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.
Compact operators and their spectrum. Compact operators. Resolvent set, spectra. Spectral theory of compact operators. Bases by eigenvectors in Hilbert 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.
Compact operators and their spectrum. Compact operators. Resolvent set, spectra. Spectral theory of compact operators. Bases by eigenvectors in Hilbert spaces.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor:
Zanco Clemente