Density of analytic polynomials in abstract Hardy spaces

Let $X$ be a separable Banach function space on the unit circle $\text{T}$ and let $H[X]$ be the abstract Hardy space built upon $X$. We show that the set of analytic polynomials is dense in $H[X]$ if the Hardy\polishendash Littlewood maximal operator is bounded on the associate space $X^{\prime }$. This result is specified to the case of variable Lebesgue spaces.