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<p>In this article, we prove that the monomials form a basis for the space of holomorphic functions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((\mathcal {H}(\ell _p), \tau _0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ℓ</mi> <mi>p</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>τ</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(p\in [1,\infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This note is motivated by the results of Dineen and Mujica in [<CitationRef CitationID="CR3">3</CitationRef>], where it was shown that the monomials form a Schauder basis for the spaces <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathcal {H}(c_0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\( \mathcal {H}_b(c_0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">H</mi> <mi>b</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, endowed with their natural topologies.</p>

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A monomial basis for the holomorphic functions on \(\ell _p\)

  • 期刊论文

  • 发表日期：

    2025年12月23日

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  • Thiago Grando,
  • Mary Lilian Lourenço

摘要

In this article, we prove that the monomials form a basis for the space of holomorphic functions \((\mathcal {H}(\ell _p), \tau _0)\) ( H ( ℓ p ) , τ 0 ) , \(p\in [1,\infty )\) p ∈ [ 1 , ∞ ) . This note is motivated by the results of Dineen and Mujica in [3], where it was shown that the monomials form a Schauder basis for the spaces \(\mathcal {H}(c_0)\) H ( c 0 ) and \( \mathcal {H}_b(c_0)\) H b ( c 0 ) , endowed with their natural topologies.