Complements of Functional Analysis
Code
11630
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Oleksiy Karlovych
Teaching language
Português
Objectives
At the end of this course students will have acquired basic knowledge and skills in the area of functional analysis in order to:
- Understand advanced contents in the area;
- Being able to start research on a topic in the area.
Subject matter
1.Topological vector spaces. Separation properties. Linear mappings. Finite-dimensional spaces. Metrization. Boundedness and continuity. Seminorms and local convexity. Quotient spaces. Examples.
2. Completeness. Baire category. The Banach-Steinhaus theorem. The open mapping theorem. The closed graph theorem. Bilinear mappings.
3. Convexity. The Hahn-Banach theorem. Weak topologies. Compact convex sets. Vector-valued integration. Holomorphic functions.
4. Duality in Banach spaces. The normed dual of a normed space. Adjoint. Compact operators.
5. Banach algebras. Complex homomorphisms. Basic properties of spectra. Symbolic calculus. The group of invertible elements.
6. Commutative Banach algebras. Ideals and homomorphisms. Gelfand transform. Involutive algebras. Positive functionals. Introduction to noncommutative algebras.
Bibliography
A. Bressan, Lecture notes on functional analysis with applications to linear partial differential equations, American Mathematical Society, 2013.
J. Cerdà, Linear functional analysis, American Mathematical Society, 2010.
E. Kaniuth, A course in commutative Banach algebras, Springer, 2009.
E.Kreyszig, Introductory functional analysis with applications, John Wiley & Sons, 1978.
W. Rudin, Functional analysis, McGraw-Hill, 1991.