Inner product space problem

Question

First of all, I apologize for my English. I'm Spanish, so I hope you can all understand me.

Here is my problem. Given the inner product:

$$ \int_0^\pi f(x)g(x)dx\ $$

in the space of continuos real valued functions, I have to calculate the angle between the vectors $ \sin(x) $ and $ \cos(x) $.

I know the formula, that is a consequence of the Cauchy–Schwarz inequality, but I am having trouble calculating the norm of the vectors.

Also, is this angle unique or varies according to the inner product? And what about the norm of a vector? Why?

Thank you!

Answer 1

Hint: if you call $\;x\;$ the wanted angle, then

$$\cos x=\frac{\langle \sin x,\cos x\rangle}{||\sin x||\,||\cos x||}$$

For example

$$||\sin x||=\left(\int\limits_0^\pi\sin^2x\,dx\right)^{1/2}=\left(\left.\frac12(x-\sin x\cos x)\right|_0^\pi\right)^{1/2}=\sqrt\frac\pi2$$

and now use the above to evaluate $\;||\cos x||\;$ (hint: it's the same value...), but the really easy value and what solves at once the whole exercise is $\;\langle\sin x,\cos x\rangle\;\ldots\ldots$

Answer 2

Well, the norm is the square root of the inner product of the vector with itself, so, for example, $|\cos x| = \sqrt{\int_0^\pi \cos^2 x d x}.$

The angle will certainly change with the inner product.