“triangle inequality”的概念、定义、习题解题方法及翻译-数学辞典

单词

triangle inequality

释义

triangle inequality

An inequality which, in different contexts, takes somewhat different forms but in each case essentially states: ‘The length of a side of a triangle is less than the sum of the lengths of the other sides.' For example, if z1 and z2 are complex numbers, then

This follows from the fact that |OQ| ≤ |OP1| + |P1Q|, where P1, P2, and Q represent z1, z2, and z1 + z2 in the complex plane.

The parallelogram OP1QP2

Likewise, for vectors a and b,

Finally, the inequality for metric spaces

is also referred to as the triangle inequality. Here a,b,c are points in a metric space and d denotes the metric.

The triangle inequality generalizes naturally to finite and infinite series. For example, and the inequality

for absolutely convergent series. A similar inequality holds for integrals

provided that f and |f| are integrable. See also reverse triangle inequality.

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