1. ## Orthonormal Set

I have this problem that says show that $\left \{ \frac12, sin(x), cos(x) \right \}$ is an orthonormal set relative to the inner product defined by $\frac1\pi \int_{-\pi}^{\pi}f(x)g(x)dx$. When I take the norms of each item in the set I get 1/2, 1, 1 respectively. Could someone point out to what I am doing wrong?

Thanks

2. Are you sure it's supposed to be an orthonormal set and not just an orthogonal set? I get what you get for the norms, so the set is definitely not orthonormal w.r.t. that inner product. Maybe they meant to have the set

$\left\{\frac{\sqrt{2}}{2},\sin(x),\cos(x)\right\},$

which I believe is orthonormal w.r.t. that inner product.

3. I went ahead and made it orthonormal and I got the same set that you derived. I am not sure if my teacher made a mistake or he wanted us to make them orthonormal. Thanks for your help!

4. You're welcome!