I am asking about the proof of corollary 2.0.2 in this: http://www-users.math.umn.edu/~garrett/m/fun/notes_2012-13/05b_banach_fourier.pdf
I understand most of the proof, but I don't understand how they proved injectivity.
They define $Tf = \{\hat{f}(n) : n \in \mathbb{Z}\} \in c_0$.
They conclude $T$ is injective because of the density of finite Fourier series in $C^0$ and density of $C^0$ in $L^1$. I don't understand how this follows at all! If someone could tell me which results they've used to conclude this, it would be much appreciated.
So the general principle is that $L^2(\mathbb{T})$ is isometrically isomorphic to $\ell^2(\mathbb{Z})$, where $\mathbb{T}$ is the torus, and the isometric isomorphism is
$$f \mapsto \{\hat{f}(n)\}_{n \in \mathbb{Z}}.$$ So, an $L^2$ function is wholly determined by its fourier coefficients. If all the fourier coefficients of $f$ are zero, then since the identification is an isomorphism, $\|f\|_{L^2(\mathbb{T})}=0$!
The isometry is encapsulated in Parseval's identity:
$$\|f\|_{L^2(\mathbb{T})}^2 = \sum_n |\hat{f}(n)|^2.$$