Question
Consider the functions e_{n}:[-\pi, \pi] \rightarrow \mathrm{C}, \quad t \mapsto e^{i n t} \text { and the vector space } V_{00}=\operatorname{span}\left\{e_{n}: n \in \mathbb{Z}\right\} \text { equipped with inner product } \langle f, g\rangle=\int_{-\pi}^{\pi} f(t) \overline{g(t)} d t, \quad f, g \in V_{00} \text {
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