A monomial basis for the holomorphic functions on certain Banach spaces

TL;DR

This paper proves that monomials form a basis for the space of holomorphic functions on specific Banach spaces, extending previous results to new classes of spaces with detailed topological and basis constructions.

Contribution

It establishes that monomials form a basis for holomorphic functions on certain Banach spaces, generalizing known results to new spaces with explicit topological frameworks.

Findings

Monomials form a basis for $ au_0$-topology on specified Banach spaces.

Constructed a fundamental system of compact sets in these spaces.

Developed seminorms generating the topology $ au_0$ in $ ext{Hol}(Z)$.

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.