Let
$$c_0(\mathbb{Z}) = \left\{(\alpha_n)_{n\in\mathbb{Z}} : \lim_{\lvert n\rvert\to\infty} \alpha_n = 0\right\}$$
be the vector space of null sequences.
Define the Fourier transform which sends every function $f\in L^1(S^1)$ to sequence of Fourier coefficients of $f$, show that Fourier transform is injective.
To solve this problem I have showed that if all the Fourier coefficients of a function $f\in L^1(S^1)$ are zero then $f=0$ a.e, then how the above defined map is injective! May be I am missing some silly points, please guide me.
