# Distance Metric Space

(Redirected from metric space)

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

## References

### 2012

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

### 2009

• (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 ...

### 1955

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