metric space

• Oct 7th 2010, 09:40 AM
nerdo
metric space

Attachment 19217
i know and understand the definition of a metric space:
Attachment 19216

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
• Oct 7th 2010, 11:17 AM
Plato
Is this what you mean?

[MATh]
\varsigma (x,y) \leqslant \varsigma (x,z) + \varsigma (z,y) [/tex]
• Oct 7th 2010, 11:28 AM
nerdo
• Oct 7th 2010, 11:45 AM
Plato
Can you explain where you are having trouble?
• Oct 7th 2010, 11:49 AM
nerdo
i don't understand how the case if x=y but not z still holds
• Oct 7th 2010, 12:30 PM
Plato
Is this true: $\displaystyle \varsigma (x,z) = \left\| x \right\|_2 + \left\| z \right\|_2 \;\& \,\varsigma (z,y) = \left\| z \right\|_2 + \left\| y \right\|_2 ?$

What if you were to add those together?

Is $\displaystyle 2\left\| z \right\|_2 \geqslant 0?$
• Oct 7th 2010, 11:29 PM
nerdo
Thanks for the help. It made my life much easier.
• Oct 8th 2010, 06:13 AM
HallsofIvy
Quote:

Originally Posted by nerdo
Thanks for the help. It made my life much easier.

Remarkable! Hey, Plato, can you make my life much easier?