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Metric Space Examples

See also: Examples of metric spaces (Wikipedia)

  1. d(x,y)>0 ∀x,y, s.t. x≠yd(x,y) > 0 ~ \forall x, y, ~ \textit{s.t.}~ x \neq y (positivity)
  2. d(x,y)=0  ⟺  x=yd(x,y) = 0 \iff x = y (equality)
  3. d(x,y)≤d(x,z)+d(z,y)d(x,y) \leq d(x,z) + d(z, y) (triangle inequality)

Normed Metric Spaces#

Normed metric spaces are induced by some norm, i.e. d(x,y)=∥x−y∥d(x,y) = \|x - y\|, where ∥.∥\|.\| can be for example, Euclidian (L2L_2), Taxicab (L1L_1), Max (L∞L_\infty), or some other norm.

These require norm as a dependency. As such, need all the algebraic properties for addition/multiplication.

More specifically, normed metric is a metric that is also translation-invariant: d(x+z,y+z)=d(x,y)d(x+z, y+z) = d(x,y).

Discrete metric#

d(x,y)={1if x≠y,0if x=yd(x,y) = \begin{cases} 1 &\text{if}\ x\neq y , \\ 0 &\text{if}\ x = y \end{cases}

Metric Spaces not Induced by Norm#

For example, consider a graph G=(E,V)G = (E,V). Define distance between two vertices x,y∈Gx,y \in G to be the length of the minimal path between these vertices. Aka Geodesic distance. This metric is not translation invariant in general (check it!).