Are the trigonometric polynomials, namely, those having the domain $[a,b] \in \mathbb{R}$, and the range $\mathbb{C}$, spanned by the orthogonal vectors $e^{inx}$? I think this is assumed in the text I'm looking at, Fourier Analysis by Stein and Shakarchi, but I don't know how to show this. I would greatly appreciate any help.
2026-03-27 15:05:04.1774623904
Trigonometric polynomials are spanned by ${e^{inx}}$
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They are, because $e^{it}-e^{-it}=2i\sin t $ and $e^{it}+e^{-it}=2\cos t$.