Lecture 1: Introductory Lecture 2 Basic Definitions

We start with the definitions of metric spaces and normed spaces. Definition 1 Let X be a set, and let d : X ×X → R ∪ {0}. The pair (X, d) is a metric space if for all x, y, z ∈ X, 1. d(x, y) = d(y, x) 2. d(x, y) = 0 ⇔ x = y 3. d(x, y) + d(y, z) ≥ d(x, z) (triangle inequality) Definition 2 A normed space is R for some finite k together with an associated mapping a → ‖a‖ from R to R ∪ {0} such that: 1. For all λ ∈ R, ‖λa‖ = |λ|‖a‖ 2. ‖u+ v‖ ≤ ‖u‖+ ‖v‖ 3. ‖u‖ = 0 if and only if a = 0 (the zero vector) Note that (R, d) where d(u, v) = ‖u − v‖ is a metric space. The following are some examples of norms: