Variation of the triangle inequality on arbitrary normed spaces

  • Thread starter Bipolarity
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  • #1
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Main Question or Discussion Point

The following inequality can easily be proved on ##ℝ## :

## ||x|-|y|| \leq |x-y| ##

I was wondering if it extends to arbitrary normed linear spaces, since I can't seem to prove it using the axioms for linear spaces. (I can however, prove it using the definition of the norm on ##ℝ## by using casework).

Suggestions? Hints?

BiP
 

Answers and Replies

  • #2
fzero
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This is called the reverse triangle inequality and can be shown by applying the regular triangle inequality to

$$ \begin{split} & \| x\| = \| (x - y) + y \|, \\
& \| y\| = \| (y - x) + x \|. \end{split}$$
 
  • #3
HallsofIvy
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The "triangle inequality" itself is part of the definition "norm".
 

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