Topology and Introduction to Functional Analysis
Code
10984
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Elvira Júlia Conceição Matias Coimbra
Weekly hours
6
Total hours
84
Teaching language
Português
Objectives
Our couse is intended to familiarize the students with the basic concepts, principles and methods of topology and functional analysis. Although the emphasis is mainly on normed linear spaces, with arbitrary dimension, some of the results are established in topological linear spaces. We study the basis of the more advanced theory of normed, Banach spaces and Hilbert spaces without which the usefulness of these spaces and their applications would be rather limited.
Prerequisites
Knowledge in Linear Algebra and Mathematical Analysis.
Subject matter
1. Metric spaces. Sequences. Cauchy sequences and convergent sequences. Complete metric spaces.
2. Topological spaces. Subspaces. relative topologie. Separability. Haurdorff spaces.
3. Continuity. Continuous mappings. Homeomorphisms. Compactness. Compact sets. Connected sets.
4. Linear normed spaces. Finite dimensional linear normed spaces. Compactnness and finite dimension. Bounded and continuous linear operators. Banach spaces.
5. Inner product spaces. Hilbert spaces. Orthonormal sequences and sets. Total orthonormal sets and sequences. Series related to Total orthonormal sets and sequences. Orthogonal complements and direct sums. Representation of functionals on Hilbert spaces( Riesz ´s theorem).
6. Some applications to the approximation theory. Approximation in Hilbert spaces.
7. Fundamental theorems in Banach spaces.
Bibliography
1. J. Dieudonné – Foundations of Modern Analysis – Academic Press
2. Eloon Lages Lima – Espaços métricos – Projecto Euclides
3. Kreysig, E. – Introductory Functional Analysis with Applications – John Wiley and Sons.
4. Machado, A.. – Topologia – Universidade Aberta.
5. Taylor, A. E. – Introduction to Functional Analysis – Wiley.
6. Yosida– Functional Analysis – Springer.
Teaching method
Theoretical issues are presented and explained in the theoretical class (3h/week). These issues are applied by students in the pratical class (3h/week).
Evaluation method
Evaluation is made by three tests along the semester or a final exam. The final classification is the weighted mean of the classification of the tests or, in alternative, the mark obtained in the final exam.