Let $C_0^\infty (\mathbb R)$ be space of smooth functions with compact support on the real line. Let $L_\mathbb C ^1 (\mathbb R) \cap L_\mathbb C ^2(\mathbb R) $ be equivalence class of 1-fold and 2-fold integrable functions. Let $\mathcal F$ denote fourier transformation.
I know $\mathcal F(C_0^\infty) \subseteq L_\mathbb C ^1 (\mathbb R) \cap L_\mathbb C ^2(\mathbb R) $ holds, but is it true that $$\mathcal F(C_0^\infty)=L_\mathbb C ^1 (\mathbb R) \cap L_\mathbb C ^2(\mathbb R) $$ ?
If not true then, is $F(C_0^\infty)$ dense in $L_\mathbb C ^1 (\mathbb R) \cap L_\mathbb C ^2(\mathbb R) $ ?
Thanks in advance.
Hint: fourier transform is isometery on the shcwartz class or, by density and extension argument, on the space $L^2(\mathbb{R})$. So the image is close. It is also dense. Hence the image will be the all space.
For the density use the following duallity fact.
$\mathscr{F}(F(-x))=f(w)$. Where $ F(w)= \mathscr{F}(f(x))$.
See the following nice book. The first pages of Chapter 2 answers your question very well.
Fourier Series and Integrals (H. Dym and H. P. McKean)