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.