# vector space and norm space

• Jan 3rd 2009, 12:46 PM
sah_mat
vector space and norm space
show that every inner product is norm.
if so why innerproduct goes to a field F while norm only goes to R.
• Jan 3rd 2009, 04:16 PM
ThePerfectHacker
Quote:

Originally Posted by sah_mat
show that every inner product is norm.
if so why innerproduct goes to a field F while norm only goes to R.

I am not exactly sure what your question is.

If $\left<~ , ~ \right> : V\times V\to \mathbb{C}$ is an inner product space then one of the conditions is that $\left< x,x\right>\geq 0$. The norm is define by $\sqrt{\left< x,x \right>}$, thus, certainly this is a function on $V\times V$ to the non-negative real numbers.
• Jan 3rd 2009, 04:51 PM
HallsofIvy
Quote:

Originally Posted by sah_mat
show that every inner product is norm.
if so why innerproduct goes to a field F while norm only goes to R.

You can't prove it- that is not true. And it probably was not what you were asked to prove. What is true is that any norm gives a norm. That is, given any inner product, we can use it to define a norm- but the norm and the inner product are obviously NOT the same.

Given any inner product, u*v, say, we can define the norm of a vector v as |v*v| where "| |" is the absolute value as defined on the field F. You are asked to show that |v*v| has the defining properties of a norm.