Topology and Introduction to Functional Analysis


The aim of the course is to provide the student with the concepts, methods and elementary techniques of GeneralTopology and Functional Analysis. The course provides the foundations of the most advanced theory of normed spaces, Banach spaces, and Hilbert spaces, and despite the special emphasis on normed linear spaces (ofarbitrary dimension), structural and fundamental results are established in general topological spaces, essentials for the student who wishes to pursue advanced studies in mathematics.

General characterization





Responsible teacher

Elvira Júlia Conceição Matias Coimbra


Weekly - 4

Total - 56

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

During the Theoretical-Practical classes, the different contents of this course will be exposed. Students will be asked to solve exercises and elaborate proofs of some of the different results presented. Any questions or doubts will be addressed during the classes, or during the weekly sessions specially programmed to this effect

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

  Metric spaces

2. Linear normed spaces

3. Bounded linear operators on linear normed spaces

4. Banach spaces

5. Inner product spaces. Hilbert spaces

6. Fundamental theorems for normed linear spaces

7. Topological spaces

8. Connection in topological spaces