A monomial basis for the holomorphic functions on certain Banach spaces

In this article, we prove that the monomials form a basis for the space of holomorphic functions $(\mathcal{H}(Z),{\tau }_0)$, where $Z$ denotes either the space $c_0\left({⨁}_{i=1}^{\mathrm{\infty }}{\mathrm{\ell }}_p^i\right)$ for some $p\in [1,\mathrm{\infty })$, or the space $d_{\ast }(w,1)$, which is the predual of the Lorentz sequence space $d(w,1)$. To achieve this, we first define a fundamental system of compact subsets in $Z$, and, based on this characterization, construct a family of seminorms that generate the topology ${\tau }_0$ in $\mathcal{H}(Z)$. The present work is motivated by the results of Dineen and Mujica in \cite{DM}, where it was shown that the monomials form a Schauder basis for the space $\mathcal{H}(c_0)$ and ${\mathcal{H}}_b(c_0)$ endowed with its natural topology.