Let $n \in \mathbb{Z_{+}}$ and show that the list below is an orthonormal list of vectors in $C[-\pi,\pi]$ in the vector space of real valued functions on $[-\pi,\pi]$ with the inner product given below.
Inner product, with $f,g$ as real-valued functions: $$<f,g>\ = \ \int_{\ -\pi}^{\pi} f(x)g(x)\ dx $$ Given list: $$(\frac{1}{\sqrt{2\pi}},\frac{sin x}{\sqrt{\pi}},\frac{sin\ (2x)}{\sqrt{\pi}},..., \frac{sin\ (mx)}{\sqrt{\pi}}, \frac{cos\ x}{\sqrt{\pi}}, \frac{cos\ (2x)}{\sqrt{\pi}},..., \frac{cos\ (mx)}{\sqrt{\pi}})$$
Here I have two questions:
- What does $C[-\pi,\pi]$ mean? Does it $C$ denote continouos?
- How does one go about proving this? (I guess that I can break it down to the following cases, $\forall i,j\ \mathbb{Z}_{+}$:) $$1. \ <sin\ (ix), sin(jx)>\\ 2. <sin\ (ix), cos(jx)> \\ 3. <cos\ (ix), cos(jx)>$$
I am unsure if I should show this via some fancy trig formula, induction or some other way.
Any help is appreciated.
Just do the integrations you suggested for the three different parts, 1, 2, and 3 as you suggested, and then use the trigonometric identities for products, you must automatically be able to do these integrations.
And yes, $C$ means continuous functions.