A monomial basis for the holomorphic functions on certain Banach spaces

Abstract

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 $c_0\left(\bigoplus^\infty_{i=1}\ell^i_p \right)$ for some $p\in [1, \infty)$, or the space $d_*(w,1)$, 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.