Distance Metric Space

(Redirected from Metric Space)
Jump to navigation Jump to search

A Distance Metric Space is a 2-tuple [[math]\displaystyle{ D,d }[/math]] consisting of a vector space [math]\displaystyle{ D }[/math] and a corresponding distance function [math]\displaystyle{ d }[/math] defined over every vector pair.




  • (Wikipedia, 2012) ⇒ http://en.wikipedia.org/wiki/Metric_space
  • http://en.wikipedia.org/wiki/Metric_space#Definition
    • A metric space is an ordered pair [math]\displaystyle{ (M,d) }[/math] where [math]\displaystyle{ M }[/math] is a set and [math]\displaystyle{ d }[/math] is a metric on [math]\displaystyle{ M }[/math], i.e., a function :[math]\displaystyle{ d \colon M \times M \rightarrow \mathbb{R} }[/math] such that for any [math]\displaystyle{ x, y, z \in M }[/math], the following holds:
      1. [math]\displaystyle{ d(x,y) \ge 0 }[/math] (non-negative),
      2. [math]\displaystyle{ d(x,y) = 0\, }[/math] iff [math]\displaystyle{ x = y\, }[/math] (identity of indiscernibles),
      3. [math]\displaystyle{ d(x,y) = d(y,x)\, }[/math] (symmetry) and
      4. [math]\displaystyle{ d(x,z) \le d(x,y) + d(y,z) }[/math] (triangle inequality) .
    • The first condition follows from the other three, since: : [math]\displaystyle{ 2d(x,y) = d(x,y) + d(y,x) \ge d(x,x) = 0. }[/math] The function [math]\displaystyle{ d }[/math] is also called distance function or simply distance. Often, [math]\displaystyle{ d }[/math] is omitted and one just writes [math]\displaystyle{ M }[/math] for a metric space if it is clear from the context what metric is used.


  • (Wordnet, 2009) ⇒ http://wordnet.princeton.edu/perl/webwn
    • a set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric and satisfies the ...



  • (Kelley, 1955) ⇒ John L Kelley. (1955). “General Topology. D. van Nostrand Company.