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    Let \phi_n : [0, 2\pi] \to \mathbb{C} be the function x \mapsto e^{i n x}. What's the easiest way to show that \{ \phi_n \} is a complete orthonormal system in the vector space of continuous periodic functions [0, 2\pi] \to \mathbb{C}, with inner product \displaystyle \left< f, g \right> = \frac{1}{2\pi} \int_{0}^{2\pi} f(x) \overline{g(x)} \, dx?

    (In particular, I need to show that if f is continuous and \left< f, \phi_n \right> = 0 for all n \in \mathbb{Z} then f = 0. I presume the question wants me to show it everywhere rather than almost everywhere, since this course doesn't use measure theory.)
 
 
 
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Updated: December 10, 2010
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