1.Euclidean space. Distance, angle, symmetries (rotation, reflection).
2.Metric spaces. Examples (R^n, l^p, (C,L^p)). Limits (sequences, functions).
3.Point set topology, open & closed sets. Completeness.
4.Vector spaces. Examples. Norms -- Holder & Minkowski inequalities, Convexity.
5.Compactness and Arzela-Ascoli theorem.
6.Contraction mapping. Lipschitz constant.
7.Bonus: Duality, weak topologies.