I really need help with this exercise:
Let $f \in L_2 (\mathbb{R})$. Show the equivalence of the following statements:
(a) $f \in H_1 (\mathbb{R})$.
(b) The function $\xi \mapsto \xi \hat{f}(\xi) \in L_2 (\mathbb{R})$.
(c) $\lim_{h \to 0} (f (x + h)-f (x)) / h$ exists in $L_2 (R)$.
I think I succeed in showing that (a) and (c) are equivalent, but I really don't know how to show that they are equivalent to (b).
Hint: The fourier transform of $i \partial_x f(x)$ is $\xi \hat{f}(\xi)$ (Think about why by pulling the derivative inside the Fourier integral, then try to justify that.). Since you know that the Fourier transform is a bijective and unitary map $L^2(\mathbb{R}) \to L^2(\mathbb{R})$, you will be able to see that $\partial_x f(x) \in L^2$ and $\xi \hat{f}(\xi) \in L^2$ are equivalent.
Always be careful when proving things for the Fourier transform in $L^2$ because the Fourier integral $\int f(x) e^{ikx}$ is only defined for some functions in a dense set in $L^2$.