# Euclidean vector space

• Apr 6th 2009, 02:50 PM
jackiemoon
Euclidean vector space
Show that for two vectors v and w of a Euclidean vector space V we have |v+w| ≤ |v|+ |w|.

Can anybody help with this please?
• Apr 6th 2009, 06:45 PM
TheEmptySet
Quote:

Originally Posted by jackiemoon
Show that for two vectors v and w of a Euclidean vector space V we have |v+w| ≤ |v|+ |w|.

Can anybody help with this please?

First you will need the cauchy-schwarz inequality $ \le |v||w|$

Now starting with the square we get

$|v+w|^2==++2 \le$
$|v|^2+|w|^2+2|v||w|=(|v|+|w|)^2$

Now we take the square root and we are done.