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DEFINITION A metric on a set ''X'' is a Function (called the ''distance function'' or simply '''distance''') ''d'' : ''X'' × ''X'' → R (where R is the set of Real Number s). For all ''x'', ''y'', ''z'' in ''X'', this function is required to satisfy the following conditions: # ''d''(''x'', ''y'') ≥ 0 (''non-negativity'') # ''d''(''x'', ''y'') = 0 if and only if ''x'' = ''y'' (''identity of indiscernibles'') # ''d''(''x'', ''y'') = ''d''(''y'', ''x'') (''symmetry'') # ''d''(''x'', ''z'') ≤ ''d''(''x'', ''y'') + ''d''(''y'', ''z'') ('' Triangle Inequality ''). A metric ''d'' on ''X'' is called Intrinsic if any two points ''x'' and ''y'' in ''X'' can be joined by a Curve with Length arbitrarily close to ''d''(''x'', ''y''). For sets on which an addition + : ''X'' × ''X'' → ''X'' is defined, we call ''d'' a translation invariant metric if d for all ''x'',''y'' and ''a'' in ''X''. If the triangular inequality is strengthened to d the metric is called Ultrametric , see below. NOTES These conditions express intuitive notions about the concept of Distance . For example, that the distance between distinct points is positive and the distance from ''x'' to ''y'' is the same as the distance from ''y'' to ''x''. The triangle inequality means that the distance traversed directly between ''x'' and ''z'', is not larger than the distance to traverse in going first from ''x'' to ''y'', and then from ''y'' to ''z''. Euclid in his Work proved that the shortest distance between two points is a line; that was the triangle inequality for his geometry. Property 1 (''d''(''x'', ''y'') ≥ 0) follows from properties 2 and 4 and does not have to be required separately. EXAMPLES
: :is a Metric defining the same Topology . (One can replace by any Summable Sequence of strictly Positive numbers.) EQUIVALENCE OF METRICS For a given set ''X'' two metrics ''d''1 and ''d''2 are called topological equivalent ('''uniformly equivalent''') if the identity mapping :id: (''X'',''d''1) → (''X'',''d''2) is a Homeomorphism ( Uniform Isomorphism ). RELATION OF NORMS AND METRICS | ||
| :''d''(''x'',''y''): | ''x''-''y'' |
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| ''d''(&alpha''x'',&alpha''y'') | &alpha''d''(''x'',''y'') (''homogenity'') |
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| :x: | ''d''(''x'',0) |