# Polynomials and the Inner Product

1. Dec 11, 2006

### neutrino

The question requires me to check whether the following formulae satisfy the properties of an inner product given the linear space of all real polynomials.
$$f(1)g(1)$$

$$\left(\int_{0}^{1}f(t)dt\right)\left(\int_{0}^{1}g(t)dt\right)$$

The properties are satisfied in both cases (at least, that's my answer), but the book says 'No'. How could this be?

Last edited: Dec 11, 2006
2. Dec 11, 2006

### arildno

Eeh, wherever did you get those integrals from????
You have been GIVEN that the formula is (f,g)=f(1)*g(1).
THAT is the formula you are asked to check whether is correct or not!

3. Dec 11, 2006

### cristo

Staff Emeritus
I think these might be two different questions, since the integrals don't follow from the first expression!!

4. Dec 11, 2006

### neutrino

Sorry...cristo's right. Edited the post.

5. Dec 11, 2006

### AKG

Show some work. What makes you think that the properties are satisfied?

6. Dec 11, 2006

### neutrino

Okay, for the first one...
Under the usual rules of multiplication...

f(1)g(1) = g(1)f(1)
(cf)(1)g(1) = c(f(1)g(1))
f(1)(g+h)(1) = f(1)g(1) + f(1)h(1)
f(1)f(1) = (f(1))2 > 0 for f(1) != 0, and of course, it's zero when f(1) = 0.

Second...
int(f)int(g) = int(g)int(f)
int(f)int(g+h) = int(f)( int(g)+int(h)) = int(f)(int(g)+int(f)int(h)
int(cf)int(g) = c int(f)int(g)
int(f)int(f) = (int(f))^2 >= 0

7. Dec 11, 2006

### mathwonk

whatb ARE the properties of an inner product? MAYBE ONE OF THEM IS SUPPOSED TO BE positive definiteness.

8. Dec 11, 2006

### neutrino

Commutativity, Distributivity, Associativity and Positivity(that's the name given in the book for the last property).

I'm afraid I don't understand. :uhh:

9. Dec 11, 2006

### NateTG

What happens to the 4th condition if:
$$f(x)=\frac{1}{2}-x$$

10. Dec 11, 2006

### AKG

What is positivity?

11. Dec 11, 2006

### neutrino

f(1)f(1) = -0.5x(-0.5) = 0.25>0 ?

12. Dec 11, 2006

### AKG

He meant in relation to the integral inner product.

13. Dec 11, 2006

### neutrino

Ah! Okay, I get it. f(x) is not zero, but the inner product <f,f> is. Therefore it does not satisfy the last condition.

Similarly f(x) = x-1 is an example where the last condition is not satisfied (for the first problem), right?

14. Dec 11, 2006

### AKG

Right. When mathwonk mentioned "positive definiteness", he was referring to the following property:

<f,f> > 0 if f is not the zero polynomial, and <f,f> = 0 if f is the zero polynomial.

It is standard to call this property "positive definiteness", although you appear to be calling it "positivity".

15. Dec 11, 2006

### neutrino

Thanks for the help, guys. :)

Not me, but Tom M.Apostol. :tongue2:

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