Basic spaces and their topology

Vector spaces: Defining properties of real and complex vector spaces. Euclidean space and \(C(I,\mathbb R)\).

Normed spaces: Definition and equivalence of norms. Supremum and different \(l_p\)-norms. The space \(BC(I,\mathbb R)\).

Metric spaces: Metrics. Normed spaces as metric spaces. The Euclidean and discrete metrics.

Balls and spheres: Open balls and spheres in metric spaces. Unit balls induced by different norms. The \(l_p\)-spaces.

Open and closed sets: Interior points, boundary points, open and closed sets.