# Prove metric space

## Homework Statement

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

## Homework Equations

Show that if $$p < 1$$ then $$d_p$$ 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 $$p \geq 1$$ the function $$d_p$$ is a metric but only uses $$p$$ in the equation $$\displaystyle \frac{1}{p} + \frac{1}{q} = 1$$. Can someone give me a hint where I should start?

## Answers and Replies

Hurkyl
Staff Emeritus
Science Advisor
Gold Member
Trying specific examples is often useful.

I still can't figure out. Can you give me more hint?