# Thread: metric space

1. ## metric space

Hi can someone please help me with the following question:

i know and understand the definition of a metric space:

I can prove easily M1 and M2 but M3 is giving me problems. Can someone please help me.

M1: e(p,p)=0 by defintion.
e(p,q)>0 if p does not equal to q

M2: e(p,q)= (lpl+lql)=(lql+lpl)=e(q,p)

M3: really need help

Thanks

nerdo

2. Is this what you mean?

$$\varsigma (x,y) \leqslant \varsigma (x,z) + \varsigma (z,y)$$

3. yes i am finding it difficult to prove this. Can you please help me.

4. Can you explain where you are having trouble?

5. i don't understand how the case if x=y but not z still holds

6. Is this true: $\varsigma (x,z) = \left\| x \right\|_2 + \left\| z \right\|_2 \;\& \,\varsigma (z,y) = \left\| z \right\|_2 + \left\| y \right\|_2 ?$

What if you were to add those together?

Is $2\left\| z \right\|_2 \geqslant 0?$

7. Thanks for the help. It made my life much easier.

8. Originally Posted by nerdo
Thanks for the help. It made my life much easier.
Remarkable! Hey, Plato, can you make my life much easier?