Let n ≥ 2 be a natural number, 1 ≤ p ≤ ∞ and X a Banach space. We prove that if X* contains λ-uniformly copies of l<Stack> p k </Stack>, then:<OrderedList> <ListItem> <ItemNumber>(i)</ItemNumber> <ItemContent> <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\cal{P}}(^{n}X)$</EquationSource> </InlineEquation> contains <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$c_{\mathbb{K}}\lambda ^{n}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>c</mi> <mrow> <mrow> <mi mathvariant="double-struck">K</mi> </mrow> </mrow> </msub> <msup> <mi>λ</mi> <mrow> <mi>n</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>-uniformly copies of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$l_{\left({{{p^{\ast}}} \over n}\right)^{\ast}}^{k}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msubsup> <mi>l</mi> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mi>p</mi> <mrow> <mo>∗</mo> </mrow> </msup> </mrow> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>∗</mo> </mrow> </msup> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation> in the case p* > n </ItemContent> </ListItem> <ListItem> <ItemNumber>(ii)</ItemNumber> <ItemContent> <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\cal{P}}(^{n}X)$</EquationSource> </InlineEquation> contains λn-uniformly copies of l<Stack> ∞ k </Stack> in the case p* ≤ n. This complete a result of S. Dineen’s from 1995. </ItemContent> </ListItem> </OrderedList>

Uniformly copies of l _P ^N in the spaces of polynomials

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发表日期：

2025年12月1日

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Dumitru Popa

摘要

Let n ≥ 2 be a natural number, 1 ≤ p ≤ ∞ and X a Banach space. We prove that if X* contains λ-uniformly copies of l _p ^k , then:

(i)

${\cal{P}}(^{n}X)$ contains $c_{\mathbb{K}}\lambda ^{n}$ $c_{K} λ^{n}$ -uniformly copies of $l_{\left({{{p^{\ast}}} \over n}\right)^{\ast}}^{k}$ $l_{{(\frac{p^{*}}{n})}^{*}}^{k}$ in the case p* > n

(ii)

${\cal{P}}(^{n}X)$ contains λⁿ-uniformly copies of l _∞ ^k in the case p* ≤ n. This complete a result of S. Dineen’s from 1995.

Uniformly copies of l P N in the spaces of polynomials

摘要

Uniformly copies of l _P ^N in the spaces of polynomials