- #1

- 62

- 0

## 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?