In the space of continuous functions on $C([0,2\pi]),$ how does one go about showing that the functions ${1\over\sqrt{2\pi}},$ ${1\over\sqrt{\pi}}\cos(nx)$ and ${1\over\sqrt{\pi}}\sin(nx)$ form a complete orthonormal system? The orthonormality is rather easy to check given that the inner product is defined by: $\langle a,b \rangle=\int\limits_{0}^{2\pi} a(x) b(x) dx.$ But how about the verification of the completeness?
Is it an idea to try and show that the sum of the normalized basis functions above is equal to the identity operator?
You can use the Stone–Weierstrass theorem. It states that an unital algebra $A$ of functions is dense in the space of all real-valued functions on an interval, iff it separates points. The last property means that $\forall x,y \in [0,2\pi] \quad \exists f \in A \quad f(x) \neq f(y)$. Take $A$ to be the set of finite linear combinations of elements of your basis, and
By Stone-Weierstrass, this algebra is dense, which means that infinite sums of it's elements span the whole space.
Alternatively, you can use the fact that this basis is the eigenbasis of Laplacian, or provides decomposition into irreducible representations of $SO(2)$, which are all known to be orthonormal and complete under suitable conditions (however, as far as I know, general proofs of this use Stone-Weierstrass anyway).