“triangle inequality”的概念、定义、翻译、参考文献-科学参考

triangle inequality

Mathematics

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

$| z_{1} + z_{2} | \leq | z_{1} | + | z_{2} | .$
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,

$| a + b | \leq | a | + | b | .$
Finally, the inequality for metric spaces

$d (a, c) \leq d (a, b) + d (b, c)$

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

$| \sum_{k = 1}^{\infty} a_{k} | \leq \sum_{k = 1}^{\infty} | a_{k} |$

for absolutely convergent series. A similar inequality holds for integrals

$| \int_{a}^{b} f (x) d x | \leq \int_{a}^{b} | f (x) | d x$

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

Computer

See inequality.

单词	triangle inequality
释义	triangle inequality Mathematics 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 $\| z_{1} + z_{2} \| \leq \| z_{1} \| + \| z_{2} \| .$ 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, $\| a + b \| \leq \| a \| + \| b \| .$ Finally, the inequality for metric spaces $d (a, c) \leq d (a, b) + d (b, c)$ 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 $\| \sum_{k = 1}^{\infty} a_{k} \| \leq \sum_{k = 1}^{\infty} \| a_{k} \|$ for absolutely convergent series. A similar inequality holds for integrals $\| \int_{a}^{b} f (x) d x \| \leq \int_{a}^{b} \| f (x) \| d x$ provided that f and \|f\| are integrable. See also reverse triangle inequality. Computer See inequality.
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