A forgotten theorem of Pełczyński: $(\lambda +)$-injective spaces need not be $\lambda $-injective—the case $\lambda \in (1,2]$

IF 0.7 3区 数学 Q2 MATHEMATICS Studia Mathematica Pub Date : 2022-01-19 DOI:10.4064/sm220119-25-6
Tomasz Kania, G. Lewicki
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引用次数: 0

Abstract

. Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional 1-injective Banach space contains a hyperplane that is (2+ ε )-injective for every ε > 0, yet is is not 2-injective and remarked in a footnote that Pe lczy´nski had proved for every λ > 1 the existence of a ( λ + ε )-injective space ( ε > 0) that is not λ injective. Unfortunately, no trace of the proof of Pe lczy´nski’s result has been preserved. In the present paper, we establish the said theorem for λ ∈ (1 , 2] by constructing an appropriate renorming of ℓ ∞ . This contrasts (at least for real scalars) with the case λ = 1 for which Lindenstrauss [Mem. Amer. Math. Soc. 48 (1964)] proved the contrary statement.
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一个被遗忘的Pełczyński定理:$(\lambda +)$-内射空间不一定是$\lambda $-内射- $\lambda \in(1,2)$的情况
Isbell和Semadeni[Trans.Amer.Math.Soc.107(1963)]证明了每个有限维1-内射Banach空间都包含一个超平面,对于每个ε>0都是(2+ε)-内射的,但不是2-内射的。不幸的是,佩尔奇·恩斯基的结果没有任何证据被保留下来。在本文中,我们通过构造ℓ ∞ . 这与Lindenstrauss[Mem.Amer.Math.Soc.48(1964)]证明相反陈述的情况λ=1形成了对比(至少对于实标量)。
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来源期刊
Studia Mathematica
Studia Mathematica 数学-数学
CiteScore
1.50
自引率
12.50%
发文量
72
审稿时长
5 months
期刊介绍: The journal publishes original papers in English, French, German and Russian, mainly in functional analysis, abstract methods of mathematical analysis and probability theory.
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