• Support PF! Buy your school textbooks, materials and every day products Here!

Prove metric space

  • #1

Homework Statement



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]

Homework Equations



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

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?
 

Answers and Replies

  • #2
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
19
Trying specific examples is often useful.
 
  • #3
I still can't figure out. Can you give me more hint?
 

Related Threads on Prove metric space

Replies
2
Views
2K
Replies
6
Views
1K
Replies
9
Views
1K
Replies
4
Views
964
Replies
0
Views
841
Replies
5
Views
10K
Replies
2
Views
3K
Replies
7
Views
4K
  • Last Post
Replies
8
Views
2K
Replies
2
Views
7K
Top