While reading Folland's Real Analysis textbook, I came across the Fourier Analysis section and became troubled by Theorem 8.35. It goes as follows:
Focusing on part $a.$ of the theorem, I am confused about the sudden change of supposition where we originally considered $f \in L^1 + L^2$, but then in $a.$ suddenly consider $f \in L^p \, (1 \leq p < \infty)$. In particular, I have not seen any mention of the Fourier transform if $f \in L^p$ for $p > 2$ (could we use Schwartz functions?). Furthermore as he begins the proof he lets $f = f_1 + f_2$ where $f_1 \in L^1$ and $f_2 \in L^2$, which to me signifies that he is still using $f \in L^1 + L^2$. The proof of $a.$ makes sense to me, but only for $f \in L^1 + L^2$, how are we suddenly also considering $f \in L^p \, (1 \leq p < \infty)$?