Metric Space

A metric space is a pair \((X, d)\) where \(X\) is a set and \(d: X \times X \rightarrow \mathbb{R}\) satisfying:

1. Positivity: \(\forall x \in X, d(x, x) = 0\)
2. Symmetry: \(\forall x, y \in X, x \neq y, d(x, y) = d(y, x) > 0\)
3. Triangle Inequality: \(\forall x, y, z \in X, d(x, y) \leq d(x, z) + d(z, y)\)

Examples

\(\ell_p\) Spaces

See \(\ell_p\) spaces

Graph Metric Spaces

Let \(G=(V, E)\) be a graph with \(n\) vertices and positive edge weights. \((V, d_G)\) is a metric space where

\[D_g(x, y) = \text{sum of edge weights along the shortest path}\]