1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
    d_p(\boldsymbol{x}, \boldsymbol{y}) := \left[ \sum^n_{i=1} |x_i
    - y_i|^p \right]^{\frac{1}{p}}, p \geq 1

    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


    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?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook