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Variation of the triangle inequality on arbitrary normed spaces

  1. Oct 12, 2013 #1
    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
     
  2. jcsd
  3. Oct 12, 2013 #2

    fzero

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    This is called the reverse triangle inequality and can be shown by applying the regular triangle inequality to

    $$ \begin{split} & \| x\| = \| (x - y) + y \|, \\
    & \| y\| = \| (y - x) + x \|. \end{split}$$
     
  4. Oct 12, 2013 #3

    HallsofIvy

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    The "triangle inequality" itself is part of the definition "norm".
     
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