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Prove metric space

  1. Mar 28, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [tex]X = \mathbb{R}^n[/tex] be equipped with the metric
    [tex]
    d_p(\boldsymbol{x}, \boldsymbol{y}) := \left[ \sum^n_{i=1} |x_i
    - y_i|^p \right]^{\frac{1}{p}}, p \geq 1
    [/tex]

    2. Relevant equations

    Show that if [tex]p < 1[/tex] then [tex]d_p[/tex] is not a metric.

    3. The attempt at a solution

    I don't know what approach I should take. The textbooks have proofs showing that when [tex]p \geq 1[/tex] the function [tex]d_p[/tex] is a metric but only uses [tex]p[/tex] in the equation [tex]\displaystyle \frac{1}{p} + \frac{1}{q} = 1[/tex]. Can someone give me a hint where I should start?
     
  2. jcsd
  3. Mar 28, 2010 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Trying specific examples is often useful.
     
  4. Mar 30, 2010 #3
    I still can't figure out. Can you give me more hint?
     
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