A monomial basis for the holomorphic functions on certain Banach spaces
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866915374125023232 |
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| author | Grando, Thiago Lourenço, Mary Lilian |
| author_facet | Grando, Thiago Lourenço, Mary Lilian |
| contents | In this article, we prove that the monomials form a basis for the space of holomorphic functions $(\mathcal{H}(Z), τ_0)$, 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 $τ_0$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05138 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A monomial basis for the holomorphic functions on certain Banach spaces Grando, Thiago Lourenço, Mary Lilian Functional Analysis Primary 46G20, secondary 32A05 In this article, we prove that the monomials form a basis for the space of holomorphic functions $(\mathcal{H}(Z), τ_0)$, 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 $τ_0$ 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. |
| title | A monomial basis for the holomorphic functions on certain Banach spaces |
| topic | Functional Analysis Primary 46G20, secondary 32A05 |
| url | https://arxiv.org/abs/2507.05138 |