$L^2(\mathbb{T})$ is the set of square integrable functions on the unit circle. This is also a hilbert space over the complex field. Now objects in this vector space are functions, but what is their domain and range? Are they only defined on the unit circle?
Also, $\{e^{inx}: n\in\mathbb{Z}\}$ is an orthonormal basis. How can I show that this is a maximal orthonormal basis?
Thanks.