Actually, the triangle inequality is one of the properties that define a metric, so it holds in any metric space. Two important cases are with and with (here we are using complex modulus, not absolute value).
There is a second triangle inequality, sometimes called the reverse triangle inequality, which also holds in any metric space and is derived from the definition of metric:
The reverse triangle inequality can be proved from the first triangle inequality, as we now show.
Let be given. For any , from the first triangle inequality we have:
and thus (using for any ):
and writing (1) with :
while writing (1) with we get:
which is the second triangle inequality.
|Date of creation||2013-03-22 12:14:49|
|Last modified on||2013-03-22 12:14:49|
|Last modified by||drini (3)|
|Defines||reverse triangle inequality|