Topology and Introduction to Functional Analysis


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.

General characterization





Responsible teacher

Elvira Júlia Conceição Matias Coimbra


Weekly - 6

Total - 84

Teaching language



Knowledge in Linear Algebra and Mathematical Analysis.


 1. Bollobás, B. (1990),  Linear Analysis, an Introductory Course, Cambridge University Press.

2.    Kreyszig, E. (1978), Introductory Functional Analysis with Applications,  New York: John Wiley&Sons.

3.  Lima, E. L. (1970), Elementos de Topologia Geral, Ao Livro Técnico, Editora da Universidade de São Paulo.

4. Sutherland, W. A. (2009), Introduction to Metric and Topological Spaces, Oxford University Press.

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 two 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.

Subject matter

1. Introduction to the TIAF Course

2. Metric spaces

3. Linear normed spaces

4. Bunded linear operators on linear normed spaces

5. Banach spaces

6. Inner product spaces. Hilbert spaces

7. Fundamental theorems for normed linear spaces

8. Topological spaces

9. Connection in topological spaces


Programs where the course is taught: