Does the following inequality hold in $L_p$ spaces?
$\|fg\|_p\leq\|f\|_p\|g\|_p$
How would I go about proving this? Do I need to apply Cauchy Schwarz?
2026-04-21 22:44:01.1776811441
Norm of pointwise product of Lp functions
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Apparently this is not true, according to link:
If $\Omega = (0,1)$ and $f(x) = g(x) = x^{-1/3}$, we have $\|fg\|_2 = \infty$ but $\|f\|_2 = \|g\|_2 < \infty$.