I'm hitting a wall with this problem, even though it seems easy. I found a similar-looking problem here, but that problem will help me only if $||g||_\infty$ is finite.
I know that, because [0, 1] is measurable with finite measure and $L^a[0, 1] \subset L^b[0, 1]$ whenever $a \geq b \geq 1$, we must have that $||f||_j < \infty$ for $1 \leq j \leq 3$ (and, on that same note, that $||g||_k < \infty$ for $1 \leq k \leq 7$). I've tried using Holder's inequality, but no matter what I do, I keep arriving at the fact that $(fg) \in L^1[0, 1]$.
Any guidance or corrections here would be greatly appreciated, since I'm still trying to grasp normed linear spaces. Thank you in advance!
$$\|fg\|_2^2 = \|f^2g^2\|_1 \le \|f^2\|_{3/2} \|g^2\|_3 = \|f\|_3^2 \|g\|_6^2 < \infty.$$