In the present paper, for a fixed element v of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\ell _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> with only a finite number of zero entries, we investigate lineability and spaceability phenomena in the cases that our sets are associated (with respect to v) either with the limit comparison test or with the standard comparison test. More precisely, we show that the set constituted of all elements where the limit comparison test fails contains, up to the zero vector, a dense linear subspace of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\ell _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> of dimension <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathfrak {c}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">c</mi> </math></EquationSource> </InlineEquation>, but not infinite-dimensional closed subspaces. However, the set where such test holds for every element only admits one-dimensional linear structures. We further show that the set formed by the sequences where the standard comparison test fails contains, up to the zero vector, the following: a <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathfrak {c}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">c</mi> </math></EquationSource> </InlineEquation>-dimensional dense linear subspace; an infinite-dimensional closed subspace. Moreover, we show that every infinite-dimensional closed subspace of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\ell _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> contains an element of the latter set. As an application of our findings, we deduce that the set which is composed of all the sequences whose generated series the root (resp. the ratio) test fails contains infinite-dimensional closed subspaces. We also retrieve several known results.

On (Non)Lineability of Some Subsets of $\ell _{1}$

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发表日期：

2025年12月20日

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Pedro Emerick,
Willian Franca

摘要

In the present paper, for a fixed element v of $\ell _1$ $ℓ_{1}$ with only a finite number of zero entries, we investigate lineability and spaceability phenomena in the cases that our sets are associated (with respect to v) either with the limit comparison test or with the standard comparison test. More precisely, we show that the set constituted of all elements where the limit comparison test fails contains, up to the zero vector, a dense linear subspace of $\ell _1$ $ℓ_{1}$ of dimension $\mathfrak {c}$ $c$ , but not infinite-dimensional closed subspaces. However, the set where such test holds for every element only admits one-dimensional linear structures. We further show that the set formed by the sequences where the standard comparison test fails contains, up to the zero vector, the following: a $\mathfrak {c}$ $c$ -dimensional dense linear subspace; an infinite-dimensional closed subspace. Moreover, we show that every infinite-dimensional closed subspace of $\ell _1$ $ℓ_{1}$ contains an element of the latter set. As an application of our findings, we deduce that the set which is composed of all the sequences whose generated series the root (resp. the ratio) test fails contains infinite-dimensional closed subspaces. We also retrieve several known results.

On (Non)Lineability of Some Subsets of \(\ell _{1}\)

摘要