Functional Analysis
Code
11697
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
9.0
Weekly hours
4
Teaching language
Português
Subject matter
1. Normed spaces
1.1. Examples of normed spaces
1.2. Finite-dimensional normed spaces
1.3. Banach spaces
2. Inner product spaces, Hilbert spaces
2.1. Inner products
2.2. Orthogonality
2.3. Orthogonal complements
2.4. Orthonormed bases in infinite dimensions
2.5. Fourier series
3. Linear operators
3.1. Bounded linear operators
3.2. The norm of a bounded linear operator
3.3. The space of bounded linear operators
3.4. Inverses of operators
4. Duality and the Hahn-Banach theorem
4.1. Dual spaces
4.2. Sublinear functional, seminorms
4.3. The Hahn-Banach theorem in separable normed spaces
4.4. The general Hahn-Banach theorem
4.5. The second dual, reflexive spaces, and dual operators
4.6. Projections and complemented subspaces
4.7. Weak convergence and weak-*convergence
5. Linear operators in Hilbert spaces
5.1. The adjoint of an operator
5.2. Normal, self-adjoint and unitary operators
6. Compact operators, Fredholm operators and index theory
6.1. Compact operators
6.2. Spectral theory of self-adjoint operators
6.3. Compact self-adjoint operators
6.4. Fredholm operators
6.5. Index
6.6. Fredholm alternative
Bibliography
- R. G. Douglas, Banach algebra techniques in operator theory, Springer, 2nd edition, 1998
- I.Gohberg, S.Goldberg, Basic operator theory, Birkhäuser, 2nd edition, 2001
- B. Rynne, M. Youngson, Análise funcional linear, IST Press, 2012 (Livro base)
- B. Rynne, M. Youngson, Linear functional analysis, Springer, 2nd edition, 2007