# Thread: Is this a Metric space?

1. ## Is this a Metric space?

X = R^2 d(x,y) = (x1 - y1)^2 + (x2 - y2)^2

Having trouble proving the 3 axioms...

2. Originally Posted by pkr
X = R^2 d(x,y) = (x1 - y1)^2 + (x2 - y2)^2

Having trouble proving the 3 axioms...
The first two axioms are trivial, so you are left to prove:

$\displaystyle d(x,y)+d(y,z) \ge d(x,z)$

but $\displaystyle d(a,b)=[\rho(a,b)]^2$ where $\displaystyle \rho$ is the usual metric on $\displaystyle \mathbb{R}^2$

So:

$\displaystyle d(x,y)+d(y,z) = [\rho(x,y)]^2+[\rho(y,z)]^2=[\rho(x,y)+\rho(y,z)]^2-2\rho(x,y)\rho(y,z)$ $\displaystyle \ge [\rho(x,y)+\rho(y,z)]^2$

....................... $\displaystyle \ge [\rho(x,z)]^2=d(x,z)$

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3. Thanks for that but what exactly is p? I've never seen that notation before, is it the same as sqrt(a^2 + b^2)?

4. Originally Posted by pkr
Thanks for that but what exactly is p? I've never seen that notation before, is it the same as sqrt(a^2 + b^2)?
It is what I said, the normal metric on $\displaystyle \mathbb{R}^2$:

$\displaystyle \rho(a,b)=\sqrt{(a_1-b_1)^2+(a_2-b_2)^2}$

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5. Ah, sorry! Thanks very much.