Polynomials and the Inner Product

[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?

 

[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!

 

[cristo]

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!

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

 

[neutrino]

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

 

[AKG]

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

 

[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

 

[mathwonk]

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

 

[neutrino]

whatb ARE the properties of an inner product?

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

MAYBE ONE OF THEM IS SUPPOSED TO BE positive definiteness.

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

 

[NateTG]

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

 

[AKG]

What is positivity?

 

[neutrino]

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

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

 

[AKG]

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

He meant in relation to the integral inner product.

 

[neutrino]

He meant in relation to the integral inner product.

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?

 

[AKG]

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?

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".

 

[neutrino]

Thanks for the help, guys. :)

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

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