I have an assignment in where i need to prove if a given continuous base is orthonormal and complete. I have the theory but no examples as a starting point.
$$\phi_n (k,x) = \Bigg\{ \sqrt{\frac{2}{\pi}} sin(kx) \Bigg\}$$
with:
$$0 \leqslant k < \infty, 0 \leqslant x < \infty$$
In the course notes I have, the test for orthonormality says:
$$\int_a^b p(x) \phi^*(k,x)\phi(k,x)dx = \delta(k - k') $$
I'm very confused cause in the given base i don't know how to do this test, by the presence of the weight function $p(x)$
$$\frac{2}{\pi}\int_0^{\infty} p(x) sin(kx) sin(kx) dx $$
from this point i assume the two "vectors" are the same so the inner product must be 1, but again i don't know what to do with the presence of the $p(x)$
Given a system of function, you have to specify with respect to which weighting function you want them to be complete and orthogonal. You also have to specify which domain you are considering.
In your case, they form a complete orhogonal system with respect to the weighting function $p(x)=1$ on the interval $\left[0,\pi\right]$ (in fact on any interval of length $\pi$).
This comes from:
$$\int_0^{\pi} sin^2\left(k x \right) = \frac{\pi}{2}$$
which precisely compensates your normalization factor.