Basic Properties of Metric and Normed Spaces

1 Definitions and Examples 1.1 Metric and Normed Spaces Definition 1.1. A metric space is a pair (X, d), where X is a set and d is a function from X ×X to R such that the following conditions hold for every x, y, z ∈ X. 1. Non-negativity: d(x, y) ≥ 0. 2. Symmetry: d(x, y) = d(y, x). 3. Triangle inequality: d(x, y) + d(y, z) ≥ d(x, y) . 4. d(x, y) = 0 if and only if x = y. Elements of X are called points of the metric space, and d is called a metric or distance function on X. Exercise 1. Prove that condition 1 follows from conditions 2-4. Occasionally, spaces that we consider will not satisfy condition 4. We will call such spaces semi-metric spaces. Definition 1.2. A space (X, d) is a semi-metric space if it satisfies conditions 1-3 and 4 ′: 4 ′. if x = y then d(x, y) = 0. Examples. Here are several examples of metric spaces. 1. Euclidean Space. Space R equipped with the Euclidean distance d(x, y) = ‖x−y‖2.