Real Analysis proof (inner product)

[malcmitch20]

Hello all,

I am having trouble showing that the operation defined by f*g(f of g)= Integral[from a to b]f(x)g(x) is an inner product.

I know it must fulfill the inner product properties, which are:
x*x>=0 for all x in V
x*x=0 iff x=0
x*y=y*x for all x,y in V
x(y+z)=x*z+y*z
(ax)y=a(xy)=x(ay)

I started the first one w/ Integral[a,b] f(x)*f(x) but I am not sure how to even integrate a function that is not defnied! Any help with this will get me going and I think I'll be able to complete the rest. Any ideas?

 

[AKG]

You don't need to calculate the integral, you just need to show it's non-negative.

 

[lavinia]

Hello all,

I am having trouble showing that the operation defined by f*g(f of g)= Integral[from a to b]f(x)g(x) is an inner product.

I know it must fulfill the inner product properties, which are:
x*x>=0 for all x in V
x*x=0 iff x=0
x*y=y*x for all x,y in V
x(y+z)=x*z+y*z
(ax)y=a(xy)=x(ay)

I started the first one w/ Integral[a,b] f(x)*f(x) but I am not sure how to even integrate a function that is not defnied! Any help with this will get me going and I think I'll be able to complete the rest. Any ideas?

None of these are hard. For instance x*x>=0 because x^2>=0. The thing you need to ask yourself is whether X^2 is integrable i.e. whether its integral is finite if x is integrable. In general this is not true and you must restrict the space of functions to those for which it is.

But how do you know that x.y is integrable just because x and y are?