Inner product space help

• May 6th 2010, 06:02 AM
MichaelChun
Inner product space help
There are two continuous function p and q in the continuous interval[-1,1] , prove the following inner product space is undefined

Integrate (x*p(x)*q(x)) d(x) (from -1 to 1)

I have no clue in how to prove this =(
• May 6th 2010, 06:15 AM
dwsmith
Quote:

Originally Posted by MichaelChun
There are two continuous function p and q in the continuous interval[-1,1] , prove the following inner product space is undefined

Integrate (x*p(x)*q(x)) d(x) (from -1 to 1)

I have no clue in how to prove this =(

$\displaystyle <p,p> \geq 0$ and 0 iff p=0
$\displaystyle <p,q>=<q,p>$
$\displaystyle <\alpha x+\beta y,z>=\alpha <x,z>+\beta <y,z>$
• May 6th 2010, 08:11 AM
HallsofIvy
Quote:

Originally Posted by MichaelChun
There are two continuous function p and q in the continuous interval[-1,1] , prove the following inner product space is undefined

Is "undefined"???

Quote:

Integrate (x*p(x)*q(x)) d(x) (from -1 to 1)

I have no clue in how to prove this =(
Surely you mean just "show that the integral defines an inner product on that space". And you do that, as dwsmith said, by showing that the "rules" for an inner product are satisfies.
• May 6th 2010, 02:36 PM
MichaelChun
Quote:

Originally Posted by HallsofIvy
Is "undefined"???

Surely you mean just "show that the integral defines an inner product on that space. And you do that, as dwsmith said, by showing that the "rules" for an inner product are satisfies.

cannot be defined, someone told me because in this case <p,p> is not a positive definite, but why?
I don't know how to show it
• May 6th 2010, 02:53 PM
dwsmith
• May 6th 2010, 03:24 PM
MichaelChun
Quote:
Nice! Which means when p is not 0 and the inner product is zero so we cannot define it as inner product?? That's smart
• May 6th 2010, 03:28 PM
dwsmith
Quote:

Originally Posted by MichaelChun
Nice! Which means when p is not 0 and the inner product is zero so we cannot define it as inner product?? That's smart

Correct, 0 iff p=0