<p>In this note, we consider the space  <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H(\Omega )^{\mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="double-struck">N</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>  of sequences of holomorphic functions on an open set <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega \subset \mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation>. If  <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>  is endowed with its natural topology and  <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H(\Omega )^{\mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="double-struck">N</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>  is endowed with the product topology, then it is proved the existence of two closed infinite-dimensional vector subspaces of  <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$H(\Omega )^{\mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="double-struck">N</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>  such that all nonzero members of the first subspace are sequences tending to zero pointwise but not compactly on  <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> and all nonzero members of the second subspace are sequences tending to zero compactly but not uniformly on  <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>. This complements the results provided in a recent work by the same authors.</p>

Spaceability of special families of null sequences of holomorphic functions

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2026年3月10日

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L. Bernal-González,
M. C. Calderón-Moreno,
J. López-Salazar,
J. A. Prado-Bassas

摘要

In this note, we consider the space $H(\Omega )^{\mathbb {N}}$ $H {(Ω)}^{N}$ of sequences of holomorphic functions on an open set $\Omega \subset \mathbb {C}$ $Ω \subset C$ . If $H(\Omega )$ $H (Ω)$ is endowed with its natural topology and $H(\Omega )^{\mathbb {N}}$ $H {(Ω)}^{N}$ is endowed with the product topology, then it is proved the existence of two closed infinite-dimensional vector subspaces of $H(\Omega )^{\mathbb {N}}$ $H {(Ω)}^{N}$ such that all nonzero members of the first subspace are sequences tending to zero pointwise but not compactly on $\Omega $ $Ω$ and all nonzero members of the second subspace are sequences tending to zero compactly but not uniformly on $\Omega $ $Ω$ . This complements the results provided in a recent work by the same authors.