In the first week, we will begin with the third section of Chapter 7 in the BG book,  which explores the concept of metric spaces and introduces several key topological ideas. Specifically, we will extend topological concepts previously covered in Mathematics II (MDA102), such as limit points, closed and open sets, and others, to the more general setting of metric spaces.

The theoretical content for the first week corresponds to paragraphs 7.3.1 to 7.3.5 of the BG book and covers the following concepts:

1) The notion of metric spaces, metrics and norms;

2)  Convergence  of sequences and the notion of Cauchy sequences;

3)  Basic notions in topology of metric spaces: Open and closed sets, limit points, etc.;

4) The notion of continuous functions on metric spaces;

5) L_p-norms, Hölder's inequality, Minkowski's inequality;

6) L_p-norms for sequences and functions.

Examples and applications related to these  concepts are provided in the corresponding left column of the BG book, particularly in Section E,  titled  ``Metric spaces'' and in Section F, titled  ``Convergence''.