Normed Spaces: Definitions and Examples

This lecture introduces normed vector spaces, focusing on the definition and properties of normed spaces. It covers the triangle inequality, homogeneity, and the concept of a norm. Examples include Euclidean norm, polynomials with sup norm, and functions with compact support. The lecture also revisits open, closed, and compact sets in normed spaces, emphasizing the importance of completeness. It explains how the existence of a norm defines convergence and introduces the notion of a Banach space. Additionally, it discusses dense subspaces, separability, and the closure of sets in normed spaces.