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

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  • $\begingroup$ Any book on Fourier series has a proof of the fact that $(e^{inx})$ is a basis. $\endgroup$ Apr 21 '20 at 0:11
  • $\begingroup$ @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... $\endgroup$ Apr 21 '20 at 0:13
  • $\begingroup$ There is another basic result which says the continuous functions are dense in $L^{2}(T)$. $\endgroup$ Apr 21 '20 at 0:15
  • $\begingroup$ @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? $\endgroup$ Apr 21 '20 at 0:17
  • $\begingroup$ See here. $\endgroup$ Apr 21 '20 at 0:49

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