Metric Space

A Metric Space is a 2-Tuple consisting of a Tuple Point Set D and a corresponding Distance Function d defined over every pair of Tuple Points.

• AKA: Vector Space, Distance Metric Space.
• Context:
  • It can be a Normed Vector Space.
    • an L1 Norm Metric Space (with an L1 Norm Distance Function).
    • an L2 Norm Metric Space (with an L2 Norm Distance Function).
    • a p Norm Metric Space.
  • It can be:
    • a Bounded Metric Space, if there is a maximum distance between two points in the space.
    • an Unbounded Metric Space, otherwise.
  • It can be associated with a Cluster.
  • It can be an Input to:
    • a Similarity Search Task.
• Example(s):
  • Euclidean Space (with [[Euclidean Distance Function).
  • Real Number Space (with Distance Function d(x,y) = |y-x|
  • Graph Space, (with Graph Distance Function of shortest path between two Graph Nodes.
• See: Space, Affine Space, Abstract Space, Vector Space, Dimension, Set Measure Space, Topology Space.

References

• 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 ...
• (Wikipedia, 2009) http://en.wikipedia.org/wiki/Normed_vector_space
• (Wikipedia, 2009) http://en.wikipedia.org/wiki/Metric_space
  • In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.
  • The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space. In fact, the notion of "metric" is a generalization of the Euclidean metric arising from the four long known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line connecting them.
  • The geometric properties of the space depends on the metric chosen, and by using a different metric we can construct interesting non-Euclidean geometries such as those used in the theory of general relativity.
  • A metric space also induces topological properties like open and closed sets which leads to the study of even more abstract topological spaces.
  • A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, that is, a function
    • d : M \times M \rightarrow \mathbb{R}, such that for any x, y and z in M
    • 1. d(x, y) ≥ 0 (non-negativity)
    • 2. d(x, y) = 0 if and only if x = y (identity of indiscernibles)
    • 3. d(x, y) = d(y, x) (symmetry)
    • 4. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).
  • The function d is also called distance function or simply distance. Often d is omitted and one just writes M for a metric space if it is clear from the context what metric is used. Relaxing the second requirement, or removing the third or fourth, leads to the concepts of a pseudometric space, a quasimetric space, or a semimetric space. If the function takes values in the extended real number line, but otherwise satisfies above conditions, then it is called an extended metric; the corresponding space is then called an \infty-metric space.
  • The first of these four conditions actually follows from the other three, since:
    • 2d(x, y) = d(x, y) + d(y, x) ≥ d(x,x) = 0.
  • Some authors require the set M to be non-empty.
• http://en.wiktionary.org/wiki/metric_space
  • Any space whose elements are points, and between any two of which a non-negative real number can be defined as the distance between the points ...

1955

• John L Kelley. (1955). "General Topology. D. van Nostrand Company.