Square integrable functions on T

$$L^2(\mathbb{T})$$ 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.

• Any book on Fourier series has a proof of the fact that $(e^{inx})$ is a basis. Apr 21 '20 at 0:11
• @KaviRamaMurthy I know its an orthanormal set but I dont know how to show it is a maximal orthanormal set. I know I can use Weierstrass to show the set is dense in the space of continuous functions, but L^2(T) contains discontinuous functions too... Apr 21 '20 at 0:13
• There is another basic result which says the continuous functions are dense in $L^{2}(T)$. Apr 21 '20 at 0:15
• @KaviRamaMurthy Oh. Do you know where I can find that result? I know very little about measure theory. Also, are the functions in L^2(T) only defined on the unit circle, or the whole complex plane? Apr 21 '20 at 0:17
• See here. Apr 21 '20 at 0:49