My question comes from my confusion in studying the Fefferman-Stein inequality which says that for any $f\in L^p$ it holds $\|f\|_p\leq c \|f^\#\|_p$ with $f^\#$ the maximal function. This is an apriori estimation by knowing $f\in L^p$, and I am wondering if the inverse holds: the inequality $\|f^\#\|_p<\infty$ implies $f\in L^p$? This recalls me similar situations like the soblev space $H^1_0(\Omega)$ where we check only the equivalent norm $\int_\Omega |\nabla f|^2<\infty$. So in which situation can we use only the norm to know if $f$ is in a Banach space $(X,\|\cdot\|)$? A counter example is $C^0(\bar\Omega)$ (for some bounded domain $\Omega\subset \mathbb R^n$) equipped with the maximum norm.