I am trying to understand the proof that the following two statements are equivalent. For fixed $R>0$ and $f\in L^p(\mathbb{R}^n)$ let $$S_Rf(x)=\int_{|\xi|<R} \hat{f}(\xi)e^{2\pi i x . \xi}\,d\xi.$$
1) For all $f\in L^p$, $S_Rf \to f$ in $L^p$ as $R\to \infty$.
2) There exists a constant $c>0$ depending only on $p$ such that $||S_Rf||_{L^p}\leq c||f||_{L^p}$ for all $f$
Basically I have read everywhere that for 1) $\implies$ 2) (this is the direction I am interested in) you use the UBP but this is what I'm having difficulty understanding! How, for fixed $R>0$, is $S_R : L^p \to L^p$ a bounded map? I don't even see how $S_Rf$ is even in $L^p$? Any help would be appreciated, I've been trying to work it out for hours now. Thanks