On some inequalities in $L_p$ spaces Let $f$ be a function such that $\|fg\|_1<\infty$ whenever $\|g\|_2<\infty$. I would like to show that $\|f\|_2<\infty$. It seems that I should use some kind of Hölder inequalities, since we have $\|fg\|_1\leq \|f\|_2\|g\|_2$, but I don't know how. Any help would be appreciated. Thanks!

You have to assume that 
$$M := \sup \{ \|f \cdot g\|_1; \|g\|_2 \leq 1\}<\infty$$
... otherwise it won't work. (Assume $M=\infty$. Then for all $n \in \mathbb{N}$ there exists $g_n \in L^2$, $\|g_n\|_2 \leq 1$, such that $\|f \cdot g_n\|_1 \geq n$. And this means that there cannot exist a constant $c$ such that $\|f \cdot g\|_1 \leq c \cdot \|g\|_2$, in particular $f \notin L^2$ (by Hölder inequality).)
