Fourier transform of the circle I'm asked to show that the map $F:L^1[0,2\pi]\to c_0(\mathbb{Z})$ sending a function to its sequence of Fourier coefficients has the properties that:
a) F is a bounded linear operator and we can compute its norm
b) F is injective with dense image
c) F is NOT surjective
Here $l^\infty(\mathbb{Z}):=\{(a_n)_{n\in\mathbb{Z}}|a_n\in\mathbb{C}, \sup\limits_{n}|a_n|<\infty\}$ is a Banach space with norm $\|(a_n)\|_\infty:=\sup\limits_n|a_n|.$
$c_0(\mathbb{Z}):=\{(a_n)\in l^\infty(\mathbb{Z})|a_n\in\mathbb{C}, \lim\limits_{n\to\infty}|a_n|=0\}$ 
and for $u\in L^1[0,2\pi]$ we define $a_n:=\int_0^{2\pi}u(x)e^{-inx}dx$ to be the fourier coefficients of $u$.
For part (a) I have that for $u\in L^1[0,2\pi]$ we have
$|a_n|=|\int_0^{2\pi}u(x)e^{-inx}dx|\leq\int_0^{2\pi}|u(x)||e^{-inx}|dx\leq\int_0^{2\pi}|u(x)|(1)dx=\|u\|_{L^1[0,2\pi]}<\infty$ 
Thus taking the supremum we have that $\|Fu\|_\infty=\sup\limits_{n}|a_n|\leq\|u\|_{L^1[0,2\pi]}$. So $F$ is bounded.But I am not sure if this is correct or how to calculate the actual norm.
The main ones I am stuck on are (b) and (c).
I think I can show injectivity for (b) but not so sure about showing that the image is dense. 
For part (c) I have defined $u_N(x):=\sum\limits_{n=0}^{2N-1}e^{inx}$ and I want to show that $\|Fu_N\|_\infty=2\pi$ for all $N$, but I wasn't sure how to show this since I'm not exactly sure how to calculate the norm.
If someone could help I would appreciate it very much.
 A: Let $f\in L^{1}(\mathbb{S})$ with $\int |f|=1$. Then 
$$
F:L^{1}\rightarrow c_0, |F|=\sup |a_{n}|=\sup |\int e^{-inx}f(x)dx|\le \sup |e^{-inx}|\int |f|=1
$$
Thus the norm of $F$ is $1$. For the rest define $G=F^{-1}$ to be
$$
G(a)(x)=\frac{1}{2\pi}\sum^{\infty}_{-\infty} a_n e^{inx}
$$
where $G(F(f))(x)$ is usually called the Fourier series of $f$. 
To show that $F$ is injective, you can approximate $f$ by a continuous function (using Lusin's theorem, say), then approximate again using a trigonometry series (using Stone-Weistrauss, say). To show it is not surjective it suffice to approximate $f$ by a $C^{2}$ function globally, then since $f$'s Fourier coefficients obey $O(n^{-2})$ order decay (using integration by parts!), a sequence like $a_{n}=n^{-1}$ would not work. Now given a sequence $l$ in $c_0$, I claim it is always possible to find another sequence $l'$ of decay rate $O(n^{-2})$ that is arbitrarily close to $c_0$ in $l^{\infty}$ topology. The details are deliberately left out for you to fill in as routine exercise. 
A good reference for Fourier analysis and some introductory real/functional analysis is Stein's analysis series, Volume I and III. Hope this helps you more than my answer at here. 
