  We show that if the Hardy-Littlewood maximal operator is bounded on a
separable Banach function space $X(\mathbb{R}^n)$ and on its associate space
$X'(\mathbb{R}^n)$, then a pseudodifferential operator $\operatorname{Op}(a)$
is bounded on $X(\mathbb{R}^n)$ whenever the symbol $a$ belongs to the
H\"ormander class $S_{\rho,\delta}^{n(\rho-1)}$ with $0<\rho\le 1$,
$0\le\delta<1$ or to the the Miyachi class
$S_{\rho,\delta}^{n(\rho-1)}(\varkappa,n)$ with $0\le\delta\le\rho\le 1$,
$0\le\delta<1$, and $\varkappa>0$. This result is applied to the case of
variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$.
