Othogonality of the trigonometric set?

Question

Let V be the complex inner product space consisting of all continuous funtion on [-pi,pi] with the inner product defined as
=the integral from -pi to pi of f times g

Now I know that it is easy to prove that the set of functions {sin(x), cos(x), sin(2x), cos(2x), ,,,} for all n natural is orthogonal because it is easy to show that the inner product
= 0 for all m,n natural numbers
= = 0 if m doesn't equal n

My question is that is this not also true for all m,n where m doesn't equal n. Why does this only work when m,n are restricted to natural numbers?

I have seen and done the proof (while trying to work with Fourier Series) and it seems to me that it should for for ALL m and n instead of just natural number.

Answer 1

no it isnt.

choos n=0.01 then the set { sin(0.01x), cos(0.01x), .. } is not orthogonal since the the integral is of cos(0.01x) * cos(0.02x) -pi,+pi is not 0.
It only works for natural numbers becauce then the integral of cosmxcosnx between -pi and +pi is 0.

Answer 2

Perhaps the reason is that the integrals, when m and n are rational or irrational, are not easy to evaluate. I think that showing a more general case for m and n is harder to prove than when m and n are natural numbers. Certainly when m and n are natural numbers, one can make use of induction to show it is true for all natural numbers. One cannot make use of induction for the more general case.