Vector Spaces and Topology

This lecture covers the concepts of vector spaces in R^n, scalar product, Cauchy-Schwartz inequality, and the topology in R^n including open balls and open and closed subsets. It also introduces methods of proof such as the pigeonhole principle. The lecture progresses to define R^n as a normed vector space, introduces the scalar product in R^n, and discusses the properties of the Euclidean norm. It concludes with the concept of closed and open sets in R^n and the demonstration methods like the pigeonhole principle.