Let $f,g \in L^2(0,1)$ My question is the following: is there a constant $C>0$ such that
$$\sup_{||f||_{2}\leq 1}||fg||_{2}\leq C ||g||_{2},$$
all I know is that we have
$$\sup_{||f||_{2}\leq 1}<f.g>\leq C ||g||_{2},$$
using cauchy schwarz inequality.
2026-04-08 19:40:41.1775677241
Inequality in a Hilbert Space: $\sup_{||f||_{2}\leq 1}||fg||_{2}\leq C ||g||_{2}$
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No, setting $f=g=\epsilon x^{-1/4}$ we have $\|f\|_2\ll 1$ for $\epsilon\ll 1$ but $$\|fg\|_{L^2} = \infty$$