If I have understood the def. of well-defined correct. I must show that
i) $F$ is a relation from $L^1$ to $C_b(\mathbb{R})$
ii) Every element in $L^1$ is related to some element in $C_b(\mathbb{R})$
iii) if $f_1$ and $f_2$ are equal in $L^1$ then is must hold that $Ff_1 = Ff_2$ in $C_b(\mathbb{R})$
I would say that by def.
$$ Ff(\phi) = \int_{- \infty}^{\infty} f(x) e^{-i\phi x} dx $$
it holds that $F$ is a relation between $L^1$ and $C_b(\mathbb{R})$, ie i), and aswell every element in $L^1$ is related to some element in $C_b(\mathbb{R})$. Since $|Ff| < \infty$ because $f$ is in $L^1$ thus ii).
Now if $f_1 = f_2 \in L^1$ then we want to compare
$$ Ff_1(\phi) = \int_{- \infty}^{\infty} f_1(x) e^{-i\phi x} dx \ \ \ \ (1) $$ and $$ Ff_2(\phi) = \int_{- \infty}^{\infty} f_2(x) e^{-i\phi x} dx \ \ \ \ (2). $$ Since $f_1 = f_2$ and obviously $e^{-i \phi x } = e^{-i \phi x } $ iii) must hold.
Am I on the right track?