# Thread: Inner product space help

1. ## 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 =(

2. 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
$\displaystyle <p,q>=<q,p>$
$\displaystyle <\alpha x+\beta y,z>=\alpha <x,z>+\beta <y,z>$

3. 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"???

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.

4. 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

5. 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

6. 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