https://en.wikipedia.org/wiki/Basis_function#Fourier_basis
It is known that 
form a basis for $L^2([0,1])$.
I want to see that $\{\sqrt2 \sin (2\pi nx+\pi x)\}, \{\sqrt 2 \cos (2\pi nx+\pi x)\}$, 1 also form a basis for $L^2([0,1])$. Assume that we already computed that these are orthonormal to each other.
(1) Is it true that those form a basis?
(2) How do I prove it? (heuristically or rigorously) Does it suffice to show that it is "complete"?
I'd appreciate any insight!
If I take the integers and add one to every element we will again obtain a copy of the integers. With this in mind, consider the input to $\{\sqrt{2}sin(2\pi nx+\pi x)\}$.