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?

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- Apr 6th 2009, 01:50 PMjackiemoonEuclidean 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, 05:45 PMTheEmptySet
First you will need the cauchy-schwarz inequality $\displaystyle <v,w> \le |v||w|$

Now starting with the square we get

$\displaystyle |v+w|^2=<v+w,v+w>=<v,v>+<w,w>+2<v,w> \le $

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

Now we take the square root and we are done.