Metric space

A metric space is a pair, $(S, d)$ of a set $S$ and a metric $d: S \times S \to \mathbb{R}_{\geq 0}$. The metric $d$ represents a distance function between pairs of points of $S$ which has the following properties:

  • Symmetry: for all $x, y \in S$, $d(x, y) = d(y, x)$
  • Non-negativity: for all $x, y \in S$, $d(x, y) \geq 0$
  • Uniqueness: for all $x, y \in S$, $d(x, y) = 0$ if and only if $x = y$
  • The Triangle Inequality: for all points $x, y, z \in S$, $d(x, y) + d(y, z) \geq d(x, z)$

Intuitively, a metric space is a generalization of the distance between two objects (where "objects" can be anything, including points, functions, graphics, or grades). The above properties follow from our notion of distance. Non-negativity stems from the idea that A cannot be closer to B than B is to itself; Uniqueness results from two objects being identical if and only if they are the same object; and the Triangle Inequality corresponds to the idea that a direct path between points A and B should be at least as short as a roundabout path that visits some point C first.

Popular metrics

  • The Euclidean metric on $\mathbb{R}^n$, with the "usual" meaning of distance which is given by $d(x,y)=\sqrt{\sum_{i=1}^n (x_i-y_i)^2}$ where $x=(x_1,x_2,\dots, x_n)$ and $y=(y_1,y_2,\dots ,y_n)$.

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