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A proof of a general property of norms

gucci

New member
Sep 1, 2013
13
So I have been asked to prove a result that is supposedly valid for any norm on any vector space. The statement to prove is: | ||x|| - ||y|| | <= ||x - y||

The problem is, I have no idea where to start with this proof. Maybe I'm missing some fundamental property of norms, but it seems that having to prove this for any norm on any vector space rules out any definitions that I might try to apply :-/

Sorry for not having anything to go on here, but I'm lost. Thank you all very much for any help you might be able to offer!
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,193
This follows from the triangle inequality. Consider the quantity $\| x-y+y \|$.