# Thread: A non translation/dilation-invariant metric

1. ## A non translation/dilation-invariant metric

Hi all,

Given a normed vector space X, the norm gives rise to a metric that satisfies
1. d(x + z, y + z) = d(x, y) and 2. d(ax, ay) = |a|d(x, y) for all x, y, z in X.

I'm looking for a vector space X with a metric that does not satisfy these. I've found that |x^3 - y^3| on R does not satisfy either. I can't think of any that satisfy 1 but not 2, and 2 but not 1.

Thanks.

2. Have you tried using the discrete metric?

$\displaystyle d(x,y)=1$ if $\displaystyle x=y$
$\displaystyle d(x,y)=0$ if $\displaystyle x\ne y$

This should satisfy (1) but not (2).

3. Originally Posted by roninpro
Have you tried using the discrete metric?

$\displaystyle d(x,y)=1$ if $\displaystyle x=y$
$\displaystyle d(x,y)=0$ if $\displaystyle x\ne y$

This should satisfy (1) but not (2).
Ah - didn't think of that. But I agree, satisfies (1) but not (2). Now for the other...

4. Normed vector space:
d(x,x)=0
d(x,y)=||x||+||y||.

5. Originally Posted by measureman
Hi all,

Given a normed vector space X, the norm gives rise to a metric that satisfies
1. d(x + z, y + z) = d(x, y) and 2. d(ax, ay) = |a|d(x, y) for all x, y, z in X.

I'm looking for a vector space X with a metric that does not satisfy these. I've found that |x^3 - y^3| on R does not satisfy either. I can't think of any that satisfy 1 but not 2, and 2 but not 1.

Thanks.
It looks as though both your questions have been answered, but as an afterthought I'd like to add that particularly important example of this is $\displaystyle \ell^{\infty}$ with the metric $\displaystyle \displaystyle d(\{x_n\},\{y_n\})=\sum_{n=1}^{\infty}\frac{|x_n-y_n|}{2^n(1+|x_n-y_n|)}$ doesn't satisfy either!