# Math Help - Verify a function is a metric...

1. ## Verify a function is a metric...

Let $(X,\| \cdot \|)$ be a normed space.

Show the function $d(x,y)= \| x - y \|$ is a metric.

2. Hi biddoggy!

Originally Posted by bigdoggy
Let $(X<\| \cdot \|)$ be a normed space.

Show the function $d(x,y)= \| x - y \|$ is a metric.
a metric satisfies:

(1) d(x,x) = 0
and $d(x,y) \not= 0$, if $x \not= y$

So: ||x-x|| = ||0|| = 0
OKAY

||x-y|| > 0, because $x \not= y$, thus $x-y \not= 0$
OKAY

(2) d(x,y) = d(y,x)

||x-y|| =||y-x||

||x-y|| = ||-(x-y)||

||x-y|| = |-1|*||x-y|

||x-y|| = ||x-y||

OKAY

(3) $d(x+z,y+z) \le d(x,y) + d(y,z)$

||x+z - (y+z)|| \le ||x+z|| + ||-(y+z)|| = ||x+z|| + ||y+z||
OKAY

Are there any questions on that?

Yours
Rapha