# Showing a Norm is not an Inner Product

1. Mar 27, 2012

### 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$_{n}$|)$^{2}$

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

2. Mar 27, 2012

### dirk_mec1

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

3. Mar 27, 2012

### 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.