Re: the norm of l^1 isn't derived from scalar product

On 23-05-2006 15:06, eugene wrote:

This is an easy question,

Indeed!

but i'd like to see other approaches, here it is:
Prove that the norm of l^1 isn't derived from scalar product

(sorry, i'm not sure whether it is correct to use exactly this
expression "derived from" in this situation).

I think that the usual expression is "induced from".

Here is what i think:
if it were so, the use Schwartz inequality:

|(x,y)| <= ||x|| * ||y|| and take a pair of vectors

x <> 0, y = 0, then it would have meant that (x,y) = 0 ->

x = y with the contradiction with our choice of x <> 0.

How do you deduce that x = y? Besides, if this argument was correct,
wouldn't it be possible to apply it to the norm of l^2?

Best regards,

Jose Carlos Santos
.

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