<p>The present study aims constructing the BK-spaces <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1243_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell _p(\mathcal {G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℓ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1243_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell _{\infty }(\mathcal {G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℓ</mi> <mi>∞</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> generated by generalized Motzkin (GM) matrix <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1243_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {G}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation>, which is conservative and derived by GM numbers. The Schauder basis of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1243_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell _p(\mathcal {G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℓ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1243_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> is obtained and the inclusion relations are presented. Additionally, the <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1243_Article_IEq6.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1243_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>- and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1243_Article_IEq8.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>-duals of <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1243_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell _p(\mathcal {G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℓ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1243_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell _{\infty }(\mathcal {G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℓ</mi> <mi>∞</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are described and finally, these spaces are examined in terms of some topological and geometric properties.</p>

Some topological and geometric properties of novel generalized Motzkin sequence spaces

期刊论文
发表日期：

2025年6月5日

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Serkan Demiriz,
Adem Şahin,
Sezer Erdem

摘要

The present study aims constructing the BK-spaces $\ell _p(\mathcal {G})$ $ℓ_{p} (G)$ and $\ell _{\infty }(\mathcal {G})$ $ℓ_{\infty} (G)$ generated by generalized Motzkin (GM) matrix $\mathcal {G}$ $G$ , which is conservative and derived by GM numbers. The Schauder basis of $\ell _p(\mathcal {G})$ $ℓ_{p} (G)$ for $1\le p<\infty $ $1 \leq p < \infty$ is obtained and the inclusion relations are presented. Additionally, the $\alpha $ $α$ -, $\beta $ $β$ - and $\gamma $ $γ$ -duals of $\ell _p(\mathcal {G})$ $ℓ_{p} (G)$ and $\ell _{\infty }(\mathcal {G})$ $ℓ_{\infty} (G)$ are described and finally, these spaces are examined in terms of some topological and geometric properties.