Basic Rules About Distance

Simple distance

These rules apply in every context where distance makes sense.

It is assumed that we have a set S of points or objects for which a distance d(x,y) is defined for every pair of points or objects x, y of the set S.

We shall not want to think of d(x,y) as a type of distance unless the following rules hold:

  1. For any x and y the distance d(x,y) is a non-negative real number.

  2. d(x, y) = 0 if and only if x = y.

  3. d(x, y) = d(y, x) for every pair x, y in S.

  4. d(x, z) <= d(x, y) + d(y, z) for every x, y, z in S.

Distance given by a norm

Sometimes, but not always, the distance function d(x,y) on a set S is constructed in terms of a norm || x || that is defined as a non-negative real number for all x in S.

In such cases the distance is given by:

d(x, y) = || x - y || .

If the distance function d is given this way, then it will satisfy the rules above for distance if the norm function satisfies:

  1. || x || >= 0

  2. || x || = 0 if and only if x = 0 .

  3. || -x || = || x || .

  4. || x + y || <= || x || + || y || .
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