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!

Do you have Hilbert space theory? Because then you can use that $T(g) = \int fg = <g,\bar{f}>$ is a bounded linear functional on $L^2$, in which case there must be an $h \in L^2$ for which $<g, \bar{f}> = <g, h>$ for all $g \in L^2$. Since $\int g(\bar{f} - h) = 0$
for all $g \in L^2$, it is not hard to show that $\bar{f} - h = 0$ a.e., so that $f = \bar{h}$ is in $L^2$.
