Showing a Norm is not an Inner Product

[Punkyc7]

Show the taxicab norm is not an IP.

taxicab norm is v=(x_{1}...x_{n})
then ||V||= |x_{1}|+...+|x_{n}|)

I was thinking about using the parallelogram law

but I would get this nasty thing(|x_{1}+w_{1}|+...+|x_{n}+w_{n}|)^{2}+(|x_{1}-w_{1}}+...+|x_{n}-w_{n}|)^{2}=2(|x_{1}|+...+|x_{n}|)^{2}+(|w_{1}|+...+|w_{n}|)^{2}Im not sure how to work with this. Am I going about this wrong?

Also there might be some typos with the absolute value signs the latex get messy

 

[dirk_mec1]

Hint: the norm doesn't satisfy linearity (it does satisfy the triangle inequality).

 

[clamtrox]

Do you mean that ||v+w||is not an inner product? This is easy to show! Just go through the definition of inner product -- one of the axioms is clearly not satisfied in this case.