Space of functions continuous on R with period $2\pi$.

Question

Define a dot product for a space of continuous on $\Bbb R$ functions with period 2$\pi$.

The first idea is to use integrals, but functions' domain is whole $\Bbb R$. Can't come up with something else.

Answer 1

Define:

$$\langle f,g\rangle = \int_0^{2\pi} f(x)g(x)dx$$

It's an inner product because if $\int_0^{2\pi} f^2=0$, then by continuity, $f=0$ on $[0,2\pi]$. By $2\pi$-periodicity, $f=0$ on $\Bbb R$.