I'm having a problem applying the closed graph theorem, which I think stems from distributions still being very new to me.
I am reading a proof in Stein and Weiss, Introduction to Fourier Analysis on Euclidean Spaces, 4.13 in Chapter 1, in the Further Results section, which begins:
Suppose for the sake of contradiction that the Fourier transform of every function $f\in L^p (\mathbb{R})$, as a tempered distribution, is a function. The closed graph theorem easily shows, for all $f\in L^p, p>2$, there is a constant $A$ so that the following holds:
$\int_{|x|\le 1} | \ \hat f(x)| \ dx \le A ||f||_p. $
I don't see how this follows from the closed graph theorem. Any help would be greatly appreciated.