# The alternate triangle inequality in normed and metric spaces?

In real and complex analysis, we used the alternate triangle inequality $\left|a-b\right| \geq \left||a|-|b|\right|$ a fair bit. I've just been introduced to the notion of real inner product spaces, normed spaces and metric spaces. Using the Cauchy-Schwarz inequality and the fact that every inner product induces a norm, I have derived the "alternate triangle inequality" for normed spaces $$\|v-w\|\geq\left|\|v\|-\|w\|\right|.$$

However, is this form of the triangle inequality used at all for normed or even general Euclidean spaces (like it was for analysis in $\mathbb R$)?

Further, is there a corresponding "alternate triangle inequality" for metrics (distance functions)?

• On the right, you should use use single bars since your value is already a real number. But to answer your question, if I recall correctly, I used this a few times in my functional analysis course in the context of general Banach spaces. Jul 26 '13 at 19:53
• @CameronWilliams Oh yes, thank you for pointing that out! Jul 26 '13 at 19:57

Yes this inequality is just as useful, and yes, the same inequality holds for metrics. You could prove (using the triangle inequality for metric spaces) that $$d(x,z)\geq|d(x,y)-d(y,z)|$$