$L^p$ Norm of product of two bounded functions

Question

If $f$ and $g$ are bounded functions in $L^p[a,b]$, does the following inequality hold in $L_p$ spaces?

$$\|fg\|_p\leq\|f\|_p\|g\|_p$$

Answer 1

Let $\varepsilon$ be smaller than $\min\{b-a,1\}$ and $f(x)=g(x):=\mathbf 1_{[a,a+\varepsilon]}(x)$. Then $$\lVert fg\rVert_p=\left(\int_{[a,b]}\mathbf 1_{[a,a+\varepsilon]}(x)\right)^{1/p}=\varepsilon^{1/p}$$ and $$\lVert f\rVert_p\lVert g\rVert_p=\varepsilon^{2/p}.$$