Functional Analysis

Code	15648
Year	2
Semester	S1
ECTS Credits	6
Workload	TP(45H)
Scientific area	Mathematics
Entry requirements	Does not have.
Learning outcomes	In this curricular unit are introduced the basic concepts of the theory of Banach and Hilbert spaces. At the end of the curricular unit the students should be able to: - apply the basic theory of complete metric spaces; - identify the main normed spaces; - apply the theory of the normed spaces and of the Banach spaces; - identify the main spaces with inner product; - apply the theory of spaces with an inner product and of Hilbert spaces; - know and use the main theorems of functional analysis: Hahn-Banach, Banach-Steinhaus, open application and closed graph; - apply the definition of spectrum of an operator; - know and use the main properties of the spectrum of an operator.
Syllabus	1. Revision of complete metric spaces 2. Normed spaces and Banach spaces 2.1 Definition, elementary properties and examples 2.2 Continuous linear operators 2.3 Functionals and dual space 2.4 Finite-dimensional Banach spaces 2.5 Compactness and Riesz Lemma 3. Inner product spaces and Hilbert spaces 3.1 Definition, elementary properties and examples 3.2 Orthogonal complement and ortogonal projections 3.3 Orthonormal sets 3.4 Functionals in Hilbert spaces 3.5 Adjoint operator 4. Fundamental theorems of Functional Analysis 4.1 Zorn’s Lemma 4.2 Hahn-Banach theorem 4.3 Dual operator 4.4 Reflexive spaces 4.5 Banach-Steinhaus theorem 4.6 Open mapping theorem and closed graph theorem 5. Spectral theory 5.1 Resolvent and spectrum of an operator 5.2 Spectral properties 5.3 Spectrum of compact operators 5.4 Spectrum of self-adjoint operators
Main Bibliography	Conway, J. B. (2013). A course in functional analysis. Springer Science & Business Media. Giles, J. R. (2000). Introduction to the analysis of normed linear spaces. Cambridge University Press. Kreyszig, E. (1978). Introductory functional analysis with applications New York: wiley. Michel, A. N., & Herget, C. J. (2009). Algebra and analysis for engineers and scientists. Springer Science & BusinessMedia. Rynne, B., & Youngson, M. A. (2011). Análise Funcional Linear. Coleção Ensino da Ciência e Tecnologia. IST Press. Taylor, A. E., & Lay, D. C. (1986). Introduction to functional analysis. Krieger Publishing
Language	Portuguese. Tutorial support is available in English.